The solution space geometry of random linear equations

The solution space geometry of random linear equations Dimitris Ac hlioptas Univ ersit y of A thens ∗ Mic hael Mollo y Univ ersit y of T oron to † Abstract W e consider rand om systems of linear equ ations ov er GF(2) in whic h ev ery equation bind s k v ariables. W e obtain a precise description of th e clustering of solutions in such systems. In particular, we prov e that with probability that tends t o 1 as the num b er of v ariables, n , grow s: for every pair of solutions σ , τ , either there exists a sequence of solutions starting at σ and ending at τ such that successiv e solutions hav e Hamming distance O (log n ), or every sequence of solutions starting at σ and ending at τ con tains a p air of successive solutions with distance Ω( n ). F u rthermore, we determine precisely which pairs of solutions are in eac h category . Key to our results is establishing the fol lowi ng high probability property of cores of random hyp ergraphs whic h is of indep endent interest. E very vertex not in the r - core of a random k -uniform hyp ergraph can b e remo ved by a sequence of O (log n ) steps, where each step amounts to remo ving one vertex of degree strictly less than r at the time of remo v al. ∗ Researc h supported b y an ER C IDEAS Starting Gran t, an NSF CAREER Award, and a Sloan F ell o wship. † Dept. of Compute r Science, Uni v ersity of T oronto . Research supp orted by an NSER C Disco v ery Gran t. 1 1 In tro duction In Random Constraint Satisfaction Pro blems (CSPs) one has a se t of n v ar iables all with t he sa me domain D and a set of independently chosen constraints, each of which binds a randomly s elected subset of k v ariable s. In the most common setting, bo th D and k a re O (1), while m = Θ( n ). Tw o canonical examples are random k -SA T and colo ring s parse rando m g raphs. A fundamental qua n tity in the study of rando m CSP s is the so-called co nstraint density , i.e., the r atio o f constra ints-to-v ariables m/n . There has been much non-r igorous evidence fr om statistical physics that for many random CSPs, if the constraint density is higher than a sp eciﬁc v alue, then all but a v anishing pr op ortion of the solutions can be partitioned int o ex po nent ially many sets (clusters) such that each set is: (i) well-separa ted, i.e., has linear Ha mming distance from all others , and (ii) in some sense, well-connected. The solution c lustering phenomenon has been a central feature of the statis tical physics approach to random CSPs and is central to impo rtant algorithmic dev elopments in the area , such as Survey P ropaga tion [19]. The mathematical studying of clustering b egan in [18, 9] where it w as shown that in ra ndom k -CNF formulas, a bove a cer tain density there exist co nstants 0 < α k < β k < 1 / 2 such that w.h.p. no pair of satisfying assig nment s has distance in the range [ α k n, β k n ]. Let us say that tw o solutions ar e adjacent if they hav e Hamming distance 1 and consider the connected components under this notion of adjacency . In [4] it was sho wn that ab ov e a certain density , ther e exist exp onentially many connected components of solutions and, moreover, in every one of them the ma jority of v ar iables are fro zen, i.e., take the same v alue in all assignments in the connected comp onent. Deﬁning a cluster-re gion to b e the union of one or mo re connected c omp onents, [3] pr ov ed that ab ov e a certain densit y , not only do exp onentially man y connected comp onents exist, but there are exp onentially many cluster-r egions separated f ro m one ano ther b y linear Hamming dista nce. Mo reov er, asymptotic bounds were given o n the volume, diameter, a nd separatio n of these clus ter regio ns. Later , in [2], it was shown for random k -SA T and r andom gra ph colo uring tha t when cluster ing o ccurs, the emerg ent cluster-reg ions ar e also separated b y la rge energetic bar riers, i.e., that an y path co nnecting solutions in diﬀerent cluster-r egions passes through v alue assignments vio lating linearly man y constraints. This picture o f cluster-regions remains unch ang ed if one co nsiders tw o solutions to be adjacent if they have Hamming distance o ( n ). A t the same time, though, it lends little information rega rding the in ternal or ganization of clus ter-regio ns, e.g., the connectivity of each s uch r egion. Un til now, it has no t b een pr ov en that any r andom CSP mo del exhibits clustering in to sets that are b oth well-separated and well-connected. The main c ontribution of this pap er is to pr ov e that this phenomenon do es indeed o ccur for random k -XOR-SA T, i.e., for systems of r andom linear equa tions over GF(2) where each equation contains precisely k v ariables. W e also obtain a precise descriptio n of the clus ters. W e remark that the c luster structure for k -XOR-SA T is m uch s impler than what is hypothesize d fo r most CSP s, e.g., the clusters ar e all isomor phic and hav e the sa me set of fr ozen variables (see Section 4 ). Random k -XOR-SA T has long b een recog nized as one of the most access ible o f the fundamental random CSP mo dels, in that resear chers ha ve managed to prove diﬃcult results for it that app ear to b e far b eyond o ur current reach for , e.g., r andom k -SA T and rando m graph color ing. F or example, the k -X OR-SA T satisﬁability threshold was established by Dubois and Mandler[11] for k = 3 and by Dietzfelbinger et al. [10] for gener al k . 1.1 Random systems of linear equations W e co nsider systems o f m = O ( n ) linear equations ov er n Bo o lean v ariables , whe re ea ch equation binds a constant n umber of v a riables. Clearly , deciding whether such a system has sa tisfying assignments (so lutions) can be do ne in p o lynomial time by , say , Gaussian elimination. In fact, the set of solutions forms a subspace, so that the sum of tw o solutio ns is also a solution. A t the sa me time, it seems that if one fails to exploit the underlying alg ebraic structure ev ery thing fa lls apart. F o r example, if the system is unsatisﬁa ble, ﬁnding a v alue as signment σ that satisﬁes a s many equa tions as p os sible, i.e., MAX X OR-SA T, is NP-complete. Moreov er, given a satisﬁable system and an a rbitrary σ ∈ { 0 , 1 } n , ﬁnding a solutio n neares t to σ is a lso NP-complete [5]. Finally , random systems of linear equa tions a ppe ar to b e extremely diﬃcult b o th for generic CSP solvers a nd for SA T solvers working o n a SA T enco ding of the ins tance. Indeed, very rece n t 2 work str ongly sugg ests that among a wide array o f rando m CSPs, ra ndom k -XOR-SA T, deﬁned b elow, is the m ost diﬃcult for random w alk type alg orithms such as W alkSat [12]. In random k - X OR-SA T, whic h we study her e, each equation binds exactly k ≥ 3 v ariables (the case k = 2 is trivial). T o form the random sys tem o f eq uations A x = b we take A to be t he adjacency matr ix of a random k -uniform hypergr aph H with n v ariables and m edges and b ∈ { 0 , 1 } m to b e a uniformly r andom v ector . It is straightforw ard to see, using e.g., Gaussian elimination, that if tw o systems ha ve the same matrix A , then their so lution s paces are isomorphic a s b ranges over v ectors for which the s olution spa ce is not e mpt y . Since we will only b e interested in prop erties of the set of solutions that a re inv a riant under isomor phism, we will assume throughout that b = 0 . As a result, thro ughout the paper we will b e a ble to iden tify the s ystem of linear equations with its underlying h yp erg raph. Reg arding the choice of random k - uniform h yp ergr aphs we will use b oth standard models H k ( n, m ) and H k ( n, p ), which respectively corresp ond to: including exactly m out of the p oss ible  n k  edges uniformly a nd indepe nden tly , and including each po ssible edge indep endently with probability p . (Results trans fer readily b etw e n the tw o mo dels when m = p  n k  .) Our corr esp onding mo dels of rando m k -XOR-SA T a re: Deﬁnition 1. X k ( n, m ) , X k ( n, p ) ar e the systems of line ar e qu ations over n b o ole an variables whose under- lying hyp er gr aphs ar e H k ( n, m ) , H k ( n, p ) and wher e we set b = 0 . The usual mo del for k -X OR-SA T diﬀers from our s only in that it takes a uniformly ra ndom Bo ole an vector b . As describ ed ab ov e, thes e mo dels are eq uiv alent up to isomor phisms of the solution space , and hence we ca n use our more conv enient deﬁnition for the purp o ses of this pap er. W e will say that a sequence of even ts E n holds with high pr ob ability (w.h.p.) for such a system if lim n →∞ Pr[ E n ] = 1 . W e will analyze X k ( n, p ). All our theor ems transla te to X k ( n, m ) where m = p  n k  using a standa rd argument. W e are interested in the ra nge p = Θ( n 1 − k ) which is equiv a lent to m = Θ( n ). W e note tha t as n → ∞ , the degrees o f the v ariables in such a r andom system tend to Poisson r andom v a riables with mea n Θ (1). This implies that w.h.p. there will b e Θ( n ) v ar iables of degree 0 and 1. Clearly , v a riables of degree 0 do not aﬀect the satisﬁability of the system. Similarly , if a v a riable v app ear s in exactly o ne equatio n e i , then we can alwa ys satisfy e i by setting v appropriately for any constant b i . Therefore, we can safely remove e i from consideratio n a nd only revisit it a fter we hav e found a s olution to the r emaining equatio ns. Crucially , this remov al of e i can cause the deg ree of other v ariables to dro p to 1. This lea ds us to the deﬁnition o f the core of a hyperg raph. Deﬁnition 2. The r -cor e of a hyp er gr aph H is the maximum sub gr aph of H in which every vertex ha s de gr e e at le ast r . It is well known [25, 21 , 15] tha t fo r every ﬁx ed r ≥ 2, a s p is incr eased, H k ( n, p ) acq uires a (mass ive) non-empty r -co re suddenly , around a critica l edge probability p = c ∗ k,r /n k − 1 . T rivia lly , removing any vertex of deg ree less than r and all its incident edges from H do es not change its r -core . Therefore, the r - core is the (p otentially empt y) outcome of the following pro cedure: rep eatedly remov e an arbitrar y v ertex of degree less than r un til no such v ertices remain. In the case of linear equations we will b e par ticularly interested in 2-co res, a s v aria bles outside the 2 -core can alwa ys b e pro p erly assigned. Deﬁnition 3. The 2-cor e system is the subsystem of line ar e quations induc e d by the 2-c or e of the underlying hyp er gr aph, i.e., the set of e quations whose variables al l lie in t he 2-c or e. A 2-cor e so lution is a solution to the 2-c or e system. An extension of a 2-c or e solut ion, σ , is a solution of the en tir e system of line ar e quations that agr e es with σ on al l 2-c or e variables. W e will show that in the absence of a 2-co re, while the diameter o f the set of solutions is linear, it is w.h.p. p os sible to tr ansform any so lution to any other solution b y changing O (log n ) v ar iables a t a time. So, the s et of s olutions is not only well-connected but pairs of so lutions exist at, esse n tially , every distance-scale. On the other hand, the emergence o f the 2-core signa ls the onset of clustering, as now every pair of solutions is either very close with re spe ct to the 2-core v ariables , or v ery fa r. 3 Theorem 1. F or every k ≥ 3 and c > c ∗ k, 2 , ther e ex ists a c onstant α = α ( c, k ) > 0 su ch that in X k ( n, p = c/n k − 1 ) , w.h.p. every p air of s olut ions either disagr e e on at le ast αn 2-c or e variables, or on at most ξ ( n ) 2-c or e variables, for any function ξ ( n ) → ∞ arbitr arily slow ly. W e will reﬁne the picture of Theor em 1, to prov e that as so on as the 2- core emerges, unless tw o solutions agree on essentially all 2-cor e v ariable s, transforming one into another r equires the simultaneous change of Ω( n ) v ar iables. T o identify the relev ant 2- core disagreements, we need to deﬁne the following notion which is central to our work. Deﬁnition 4. A ﬂippable cycle in a hyp er gr aph H is a set of vertic es S = { v 1 , . . . , v t } , wher e t ≥ 2 , wher e the s et of e dges incident to S c an b e or der e d as e 1 , . . . e t such that e ach vertex v i lies in e i and in e i +1 and in no other e dges of H ( addition mo d t ). Remark 5. Note that the vertic es v 1 , . . . , v t must have de gr e e exactly two in t he hyp er gr aph. The r emaining vertic es in e dges e 1 , . . . , e t c an have arbitr ary de gr e e and ar e not p art of the ﬂipp able cycle. Deﬁnition 6. A co re ﬂippa ble cycle in a hyp er gr aph H is a ﬂipp able cycle in the subhyp er gr aph H 0 ⊆ H induc e d by the 2-c or e of H . Thu s, in a cor e ﬂippable c ycle, the v ertices v 1 , . . . , v t hav e degree ex actly tw o in the 2-c or e , but p ossibly higher degree in H . Note also that H may contain ﬂippable cycles outside the 2-cor e. W e will prov e (Lemma 35(b)) that w.h.p. the core ﬂippable cy cles ar e disjoint. As discussed above, any 2-cor e solutio n can b e readily extended to the r emaining v a riables. Indeed, this can typically b e done in numerous wa ys since the equations no t in the 2-co re are far le ss constra ined, e.g., a constant fra ction of the eq uations outside the 2 -core form hypertrees very loo sely attached to the 2-co re. In order to understa nd the emergence of the clustering of solutio ns, we will fo cus on whether we can change the v alue of a 2-cor e v ar iable without changing many other 2 -core v ariables. If σ is any 2-cor e solution then ﬂipping the v alue of a ll v ariables in a core ﬂippable cycle r eadily yields another so lution of the 2-core, since every equation co nt ains e ither zero o r tw o of the ﬂipp ed v a riables. It is not hard to show that a rando m hyperg raph often contains a handful of short core ﬂippable cycles, implying that 2-co re s olutions may have Hamming distance Θ(1 ). At the sa me time, though, we will see (Lemma 35) that, for any ξ ( n ) → ∞ a rbitrarily slowly , the total num ber of vertices in core ﬂippable cycles w.h.p. do es not exceed ξ ( n ), placing a corr esp onding upper b ound on the distanc e betw een core solutions that diﬀer only on ﬂippable cycles. In contrast, we will pr ov e that w.h.p. every pair of co re solutions that diﬀer on even one 2- core v a riable not in a ﬂippable cycle, diﬀer in a t leas t Ω( n ) 2-core v ar iables. In other words, ﬂipping the handful of v ar iables in po ten tial ﬂippable cy cles, w.h.p. is the only kind of mo vemen t betw een 2- core solutions that does not en tail the simultaneous change of a mas sive num b er of v ariables. The ab ove indicates that the follo wing is the appropr iate deﬁnition of clusters in random k -X OR-SA T. Deﬁnition 7. Two s olut ions ar e cycle-equiv alent if o n the 2-c or e they di ﬀer only on variable s in c or e ﬂipp able cycles (while they may diﬀer arbitr arily on variables not in the 2-c or e). Deﬁnition 8. The s olution clusters of X k ( n, p = c/n k − 1 ) ar e the cycle-e qu ivalenc e cla sses, i.e., two sol ut ions ar e in the same cluster iﬀ they ar e cycle-e quivalent. Note that in the absence of a 2-cor e, this deﬁnition states that all solutions are in the same cluster. W e can now state our ma in theo rems in terms of c onnectivity prop erties of clusters. Deﬁnition 9. Two solutions σ, τ of a CSP ar e d -connected if ther e exists a se quenc e of solutions σ, σ ′ , . . . , τ such that the Hamming distanc e of every two su c c essive elements in the se quenc e is at most d . A set S of solutions is d -connected if every p air σ , τ ∈ S is d -c onne cte d. Two solution sets S, S ′ ar e d -separated if every p air σ ∈ S, τ ∈ S ′ is not d -c onne cte d. 4 It a ppea rs that for many ra ndom CSP’s, there is a co nstant α > 0 a nd a function g ( n ) = o ( n ) such that if the constra int density is suﬃciently la rge, then all but a v anishing prop or tion of the solutions can b e partitioned in to clusters S 1 , . . . , S t such that: • Every S i is g ( n )-connected. • Every pair S i , S j is αn -separated. That is the sense in whic h w e said ear lier that eac h cluster is w ell-connected and that each pair of clus ters is well-separated. Our main theorems are that for k -X OR-SA T, the clus ters we deﬁned in Deﬁnition 8 satisfy these condi- tions with g ( n ) = O (log n ). Note that for this particular CSP , the clusters contain al l the solutions, rather than all but a v anishing prop ortion o f them. Theorem 2. F or any c onstant c 6 = c ∗ k, 2 and k ≥ 3 , ther e exists a c onstant α = α ( c, k ) > 0 such that in X k ( n, p = c/n k − 1 ) , w.h.p. every p air of clusters is αn - sep ar ate d. In star k contrast, we prov e that clusters are internally very well connected. Theorem 3. F or any c onstant c 6 = c ∗ k, 2 and k ≥ 3 , ther e exists a c onstant Q = Q ( c, k ) > 0 such t hat in X k ( n, p = c/n k − 1 ) , w.h.p. every cluster is Q log n -c onne cte d. Theorem 3 is nearly tight due to the following. Observ ation 10. W.h.p. every cluster c ontains a p air of solutions that ar e not g ( n ) -c onne ct e d, for some g ( n ) = Ω(log n/ log log n ) . Pr o of. Consider an y solution σ to the 2-core, and consider an y t wo extens ions σ 0 , σ 1 of σ to the entire sy stem such that, fo r some no n-core v a riable v , we hav e σ 0 ( v ) = 0 but σ 1 ( v ) = 1. Then σ 0 , σ 1 m ust diﬀer in a t least o ne additional v aria ble in every equation containing v implying that their Hamming dista nce is at least deg( v ) + 1. If T is an acy clic (tree) comp onent o f the under lying hype rgraph a nd v is any v ertex in T , then, clearly , σ can be extended so that v takes any desired v a lue. Therefore, the maximum degree of any vertex in a tree comp onent is a lower bound for g ( n ). A tree co mpo nent T is a d -s tar if precisely one vertex in T has degr ee d and all other vertices ha ve deg ree 1. Computing the seco nd mo men t of the n umber of d -stars in a r andom hypergra ph implies that w.h.p. there e xist g ( n )-stars , where g ( n ) = Ω(log n/ lo g log n ). So, in a nutshell, we prove that before the 2-c ore emerg es a n y solution can be transfor med to any other solution a long a s equence of successive solutions diﬀering in O (log n ) v aria bles. In co nt ra st, after the 2-core emerges, the se t of solutio ns shatters into clus ters deﬁned by complete agreement on the 2-co re, ex cept for the handful o f v ar iables in core ﬂippable cycles: any tw o so lutions that disa gree on even one 2-cor e v ariable not in a core ﬂippable cycle, m ust disagre e on Ω( n ) v ariables. At the s ame time, solutions in the same cluster behave like solutions in the pre-cor e regime, i.e., one can trav el ar bitrarily inside e ach cluster by c hanging O (log n ) v a riables at a time. Our pr o of of Theo rem 3 is algo rithmic, g iving a n eﬃcien t metho d to trav el b etw een any pair of s olutions in the same cluster. Indeed, to prove Theo rem 3, we dra w hea vily from the linear structure of the constra int s to: (1) identif y a set B of fr e e variables such that the 2 | B | solutions in a ny cluster are determined by the 2 | B | assignments to B , (2) prov e that we can change these free v aria bles one-at-a -time, each time o btaining a new solution b y c hang ing only O (lo g n ) o ther v ariables. F or c < c ∗ k, 2 , there is no 2- core, and so all solutions b elong to th e same cluster. F or c > c ∗ k, 2 , but below the k -XOR-SA T satisﬁability threshold, the num ber of 2-core v ar iables ex ceeds the n umber o f 2 -core equations by Θ( n ) (see [11 , 10]), and the n um b er of v a riables on core ﬂippable cycles has exp ectation O (1 ) (Lemma 3 5); it follows tha t w.h.p. there are a n exp onential num be r of clusters. So Theorems 2, 3 yield: Corollary 11. F or every k ≥ 3 and c b elow the k -X OR-S A T satisﬁability thr eshold: 5 • If c < c ∗ k, 2 , then w.h.p. the ent ir e solution-set of X k ( n, p = c/n k − 1 ) is O (log n ) -c onne cte d. • If c > c ∗ k, 2 , then w.h.p. the solution-set of X k ( n, p = c/ n k − 1 ) c onsists of an exp onential numb er of Θ( n ) -sep ar ate d, O (log n ) -c onne cte d clusters. F or k ≥ 3, the threshold for the app ea rance of a no n-empty 2-core was determined in [21, 15] (see als o [8]) to b e: c ∗ k, 2 = min λ> 0 ( k − 1)! λ (1 − e − λ ) k − 1 . F or example, when k = 3 , an exp onential num b er of clusters emerge at c = 0 . 13 ... while the sa tisﬁability threshold[11] is at c = 0 . 15 ... . (These v alues co rresp ond to m/n = 0 . 818 ... a nd m/n = 0 . 917 ... in the X k ( n, m ) mo del.) It is not clear what happ ens, in terms of clustering, at density c = c ∗ k, 2 ; see the r emarks following Theorem 5. Our pro o f of Theorem 2 easily extends to all uniquely extendable CSPs . Deﬁnition 12 ([7]) . A c onstr aint of arity k is uniquely ex tendable if for every set of k − 1 variables and every value assignment to those variables ther e is pr e cisely one value for the unassigne d variabl e that satisﬁes the c onstr aint. Linear equations over GF( 2) and unique games are t he t wo most common examples of uniquely extendable (UE) CSP s, but many o thers ex ist (see, eg. [7]). Clear ly , any instance of a UE CSP Φ is satisﬁa ble iﬀ its 2-core is sa tisﬁable. Thus, it is natural to deﬁne clusters ana logously to XOR-SA T, i.e., tw o solutions ar e in the s ame cluster if a nd only if their 2-co re restric tions diﬀer only on core ﬂippable cycles. Our pro of o f Theorem 2 applies rea dily to any UE CSP , y ielding a corr esp onding theor em, i.e., that there exists α > 0 such that if tw o so lutions a re no t cycle- equiv alent they ar e not αn -connected (see the remark following Prop ositio n 4 8). Ho wev er, we do not know whether the analogue of Theorem 3 holds under this deﬁnition of clusters, i.e., whether it is p os sible to trav el b etw een cycle-equiv alent solutions in small steps. Also, no te that while in XOR-SA T changing all the v ariables in any ﬂippable cycle results in another so lution, this is not necessarily the case for every UE CSP Φ. Finally , we note that Theorem 1 follows immediately fro m Theo rem 2 and the fact that w.h.p. there are few er than ξ ( n ) vertices on ﬂippable cy cles (Lemma 35). Indeed, if tw o s olutions diﬀer o n more than ξ ( n ) 2 -core v ariable s, then they disagree o n a v ar iable that is not on a ﬂippable cycle. Th us, they a re in diﬀerent clusters and so disa gree on at leas t αn v ar iables, by Theor em 2. So the pap er fo cuses on pr oving Theorems 2 and 3. Remark: Ibrahimi, Kanor ia, K raning and Mon tanar i [13] hav e, independently , o btained similar results to our s. T heir deﬁnition of clus ters is equiv alent to ours, and they prov e that the clusters are w ell-co nnected and well-separa ted. Their cluster separa tion result is equiv ale n t to our Theo rem 2, but uses a diﬀerent techn ique. Their internal connectiv it y result diﬀers fr om our Theor em 3 in that (a) they pr ov e that the clusters ar e polylo g( n )-connected ra ther than O (lo g n )-connected, and (b) they a dditionally prove that the clusters exhibit a for m of high conductance. Again, their approach is diﬀerent from the o ne used her e. T o prov e high conductance, they sho w that w.h.p. the solution spac e con tains a basis whic h is polylog ( n )-sparse, meaning that each vector in the basis has Hamming distance a t most p olylo g( n ) from 0 . It is easy to see that the set of free v ariable s B that we cho ose in Section 4 yields a O (log n )-spa rse ba sis. So our pro of, along with Lemma 1 .1 of [13] combine to yield a stro nger conducta nce res ult by replacing “(log n ) C ” with “ O (log n )” in their Theorem 1. 1.2 Cores of hypergraphs The main step in o ur pro of of Theorem 3 is to prov e a prop erty of the non-2 -core vertices in a ra ndom hypergra ph. As this prop erty is of indep endent int eres t, we prov e it for non- r -c ore vertices for gener al r ≥ 2. 6 F or a ny in tegers k ≥ 2 , r ≥ 2 s uch that r + k > 4, the threshold for the appea rance of a non-empt y r -cor e in a k -unifor m rando m hyperg raph was determined in [21, 15] to b e: c ∗ k,r = min λ> 0 ( k − 1)! λ  e − λ P ∞ i = r − 1 λ i /i !  k − 1 . (1) F or r = k = 2 , i.e., for cycles in gr aphs, the emerg ence of a 2-core is trivial as any c onstant-sized cycle has non-zero probability for all c > 0. On the other hand, for r + k > 4 , any r -cor e has linear size w.h.p. The threshold for the emer gence of a 2 -core of linear size in a random gr aph coincides with the thresho ld for the emergence of a gia nt component [24], so we set c ∗ 2 , 2 = 1, consistent with the expr ession above after repla cing min with inf . Recall that we can r each the r -cor e o f a h yp erg raph by rep eatedly removing any one vertex of degr ee less than r , until no s uch vertices r emain. Consider a vertex v not in the r -core, a nd consider the goal of r ep eatedly re moving vertices of degree less than r until v is remov ed. W e prov e that w.h.p. for every non- r -core v ariable v , this c an b e achiev ed by removing only O (log n ) vertices. Deﬁnition 13. An r -stripping sequence is a se quenc e of vertic es that c an b e delete d fr om a hyp er gr aph, one- at-a-time, along with their incident hyp er e dges such that at the t ime of deletion e ach vertex has de gr e e less than r . A terminal r -stripping se quence is one t hat c ontains al l vertic es outside t he r -c or e; i.e., a se quenc e whose deletion le aves t he r -c or e. Deﬁnition 14. F or any vertex v not in the r - c or e, the depth of v is the length of a shortest r -st ripping se qu en c e ending with v . Theorem 4. F or any inte gers k ≥ 2 , r ≥ 2 and any c onst ant c 6 = c ∗ k,r , let H = H k ( n, p = c/n k − 1 ) . Ther e exists a c onstant Q = Q ( c, k , r ) > 0 such that w.h.p., every vertex v in H has depth at most Q log n . It is easy to show using standard facts ab out r -cor es of random hyperg raphs tha t for every cons tant ǫ > 0, there is a constant T = T ( ǫ ) such that w.h.p. all but ǫn of the non-core vertices have de pth at mo st T . The c halleng e here is to prov e that w.h.p. al l non-cor e vertices hav e depth O (log n ). Remark 15. The case k = r = 2, i.e. the 2- core of a ra ndom graph, follows ea sily from previous ly known work. The conclus ion of Theorem 4 do es not hold at c = c ∗ 2 , 2 = 1. (See the remarks following the sta temen t of Theorem 5 b elow.) 2 Related w ork T o get an upper bound on the random k -XOR-SA T satisﬁabilit y thr eshold, observe that the expected n umber of so lutions in a rando m instance with n v aria bles and m constraints is bounded by 2 n (1 / 2) m → 0 if m/n > 1. As one can ima gine, this condition is not tight s ince v ariables of degr ee 0 a nd 1 only co ntribute ﬁctitious degrees of fre edom. Perhaps the next simplest nec essary condition for satisﬁability is m c /n c = γ c ≤ 1, where n c , m c is the num be r of v ariables and equations in the 2-core, res pectively . In [11] Dub ois and Ma ndler prov ed that, for k = 3, this s imple necessary conditio n fo r satisﬁability is also suﬃcient b y proving that for all γ c < 1, the num be r of core solutions is str ongly concentrated around its (expo nential) expecta tion. Thus, they determined the satisﬁability threshold for 3-XOR-SA T. Dietzfelbinger et al. [10] mo dify and e xtend the approach of [11] to determine the sa tisﬁability threshold for gener al k . A full version of [11] has no t b een published, but a pro of for all k ≥ 3 app ears in [10]. M´ ezard et al. [20] w ere the ﬁrst to study clustering in ra ndom k -XOR-SA T. Sp eciﬁcally , they deﬁned the clusters b y saying that tw o solutions a re in the same cluster iﬀ they agre e o n al l v ar iables in the 2-core. They prov ed that there exists a constant γ > 0 such that for any θ ∈ (0 , γ ) and any integer z = θ n + o ( n ), w.h.p. no tw o solutions diﬀer on ex actly z v a riables in the 2- core. Based on this fact, they claimed that the clusters they deﬁned are Ω( n )-se parated, i.e., that every pair o f solutions in diﬀerent clusters is not γ n -connected. As we hav e alr eady seen, this is fa lse since it do es not acco unt for the eﬀect of co re ﬂippable 7 cycles. Performing the analysis of solutions that diﬀer on o ( n ) v ar iables is what allows us to establis h that 2-core solutions which diﬀer on o ( n ) v a riables must diﬀer only on core ﬂippable cycles. Indeed, this is the most diﬃcult par t of our pro o f of Theorem 2. M´ ezard et al. [20] als o g av e a heur istic ar gument that if σ is any solution and v is a non-core v ar iable, then ther e exists a solution σ ′ in which v takes the o ppo site v alue from the o ne in σ such that the distance betw een σ and σ ′ is O (1). F rom this they c oncluded that clusters a re well-connected. Reg arding internal connectivity , the clusters o f [20] ar e, indee d, well-connected, i.e., the analogue of Theorem 3 holds for them, since they are s ubsets of the cluster s deﬁned in this pape r. How ever their pr o of of this fact is ﬂaw ed; it imp lies that their cluster s ar e O (1)-connected, which is no t tr ue by the same argument used as for Observ ation 10. Proving that the k -XOR-SA T cluster s ar e well-connected was later listed as an ope n pr oblem in [17]. Finally , as describ ed abov e, Ib ra himi, K anoria , Kraning a nd Montanari [13] hav e, independently , obtained similar results to ours. 3 Pro of Outline 3.1 Theorem 2: Cluster separation Given a so lution σ , a ﬂipp able set is a se t o f v a riables S such that ﬂipping the v alue of all v ar iables in S yields another solution τ . P roving Theorem 2 b oils down to proving that w.h.p., in the subsystem induced by the 2-c ore, every ﬂippable set other than a ﬂippable cycle has linear size. A common approach to pr oving ana logous statemen ts is to establish that every ﬂippable set, other than a ﬂippable cycle, mu st deterministically induce a dense subgraph. In pa rticular, if o ne can prov e that for some constant ǫ > 0, every such set is at least 1 + ǫ times as dense as a ﬂippable cy cle, then standa rd ar guments yield the desired conclusion. Here, thoug h, this is not the case, due to the possibility of a rbitrarily long paths of degr ee 2 vertices. Sp eciﬁcally , by replacing the edg es of a n y ﬂippable set (that is not a ﬂippable cycle) by 2-linke d p aths , one c an eas ily create ﬂippable sets whose dens it y is arbitr arily close to that of a ﬂippable cycle (for a mo re more precise s tatement , see the deﬁnition of 2-linke d p aths in Section 9). Thus, controlling the num b er and int era ctions of these 2-linked paths, an a pproach similar to that of [1 , 2 3], is crucia l to our argument. In o rder to work o n the 2 -core, we carr y this ana lysis out o n h yp erg raphs with a given degree sequence. The key to controlling 2-linked paths is to b ound a parameter governing the deg ree to which they tend to bra nch. Le mma 32 s hows that this parameter is b ounded b elow 1, so while arbitrar ily long 2- linked paths will occur , their frequency decreases exp onentially with their length. W e no te that if we w ere working on hyperg raphs with minimum degre e at least 3, then there would b e no 2- linked paths, a nd the pr o of would hav e been very ea sy . All of the diﬃculties arise from the problem of degree 2 vertices. W e note that our approa ch applies to gener al degre e sequences of minimum degr ee 2. 3.2 Theorem 3: Connectivity inside clusters The main step in the pro of of Theo rem 3 is to prov e Theorem 4; i.e. that every vertex outside the r - core can b e remov ed by a n r -stripping sequence of length O (lo g n ). It is often useful to consider stripping the vertices in s everal parallel rounds. Deﬁnition 16. The par allel r - stripping pro cess c onsists of iter atively r emoving all vertic es of de gr e e less than r at onc e along with any hyp er e dges c ont aining any of t hose vertic es, u ntil no vertic es of de gr e e less than r r emain. T o prov e that all non-core vertices can b e r emov ed by a stripping sequence of length O (log n ), our approach is signiﬁcantly diﬀerent below and ab ov e the thr eshold, c ∗ k,r , for the emergence of an r -cor e in random k - uniform h ype rgraphs . In b oth cases, we b egin by stripping down to H B , the h yp ergra ph remaining after B rounds of the para llel stripping pro cess, for a suﬃciently large constant B . A simple ar gument shows that for a ny non-c ore vertex v , the num ber of vertices remov ed during this initial phase that ar e relev ant to 8 the remov al of v , is b ounded. Thus, what remains is to show that an y non- core v ertex in H B can b e removed from H B by a s tripping seq uence of length O (log n ). F or c < c ∗ k,r , we prov e t hat there exists a suﬃciently large constant B = B ( c, k , r ) such that all connected comp onents of H B hav e size a t most W = O (lo g n ); therefor e, any remaining vertex can be removed with an additional W s trips. T o do this w e establish a nalytic expansions for the degr ee se quence of H B as B grows and then apply a hyper graph extension of the main result of Molloy and Reed [22] regar ding the comp onent sizes o f a random k -uniform hypergr aph with a g iven degr ee se quence. F or c > c ∗ k,r , a lo t mo re w or k is requir ed. Once again, 2-linked paths are a ma jor pro blem. Indeed, it is not hard to see that a lo ng 2- linked path with o ne endpoint of degree 1, can create a long stripping sequence leading to the remov a l of its other endp oint. W e ﬁrst establish that for any ǫ > 0, there exists a suﬃcien tly lar ge co nstant B = B ( c, k , r, ǫ ) such that H B is suﬃciently close to the r -core for tw o impo rtant prop erties to hold in H B : (i) ther e are at most ǫ n vertices of degree less than r , and (ii) the “branching” parameter for 2-linked paths, mentioned ab ov e, is bo unded below 1. Prop erty (ii) allows us to control long 2-linked paths. How ever, this do es no t suﬃce a s we need to c ontrol, more generally , for lar ge tree -like stripping sequences. T o do so, w e no te that an y larg e tree must either have many leaves, o r long paths of deg ree 2 vertices. Such long pa ths will c orresp ond to 2-linked pa ths in the random hypergr aph, and so (ii) allows us to co nt ro l the latter case . Leav es of the tree will ha ve degree le ss than r , and so (i) enables us to con trol the former case. 4 An Algorithm for T ra ve ling Inside Clusters In this section, we show ho w we use Theor em 4 to prov e Theorem 3. In fact, we require Theor em 5 b elow, which is so mewhat stro nger than Theorem 4 . Given a h yp erg raph H , we co nsider any terminal r - stripping sequence, v 1 , . . . , v t , i.e., one that removes every vertex outside of the r -c ore of H . Le t H i denote the hypergra ph remaining after remo ving v 1 , . . . , v i − 1 ; so H 1 = H a nd H t +1 is the r -core of H . Let E i denote the set of at most r − 1 hyperedg es in H i that contain v i . W e form a directed graph, D , as follows: Deﬁnition 17 . The vertic es of D ar e the non- r -c or e vertic es v 1 , . . . , v t , as wel l as any r - c or e vertex that shar es a hyp er e dge with a vertex not in the r -c or e. F or e ach vertex v i in the s tripping se quenc e, D c ontains a dir e cte d ar c ( u, v i ) for every vertex u 6 = v i c ontaine d in the hyp er e dges of E i . Note that if v i has de gr e e zer o in H i , then E i = ∅ , and so v i wil l have inde gr e e zer o in D . F or every vertex v in D , we de ﬁne R + ( v ) to b e the set of vertices that can be reached fro m v . Note that if v is not in the r -co re, then the vertices of R + ( v ) can b e arra nged into a (not necess arily terminal) r -s tripping seq uence ending with v . So to prov e Theorem 4, it suﬃces to show | R + ( v ) | = O (log n ) for every such v . Theorem 5. F or any inte gers k ≥ 2 , r ≥ 2 and any c onstant c > 0 , c 6 = c ∗ k,r , let H = H k ( n, p = c/n k − 1 ) . Ther e ex ists a c onstant Q = Q ( k, r, c ) > 0 such that w.h.p. ther e is a terminal r - stripping se qu enc e of H for which in t he digr aph D asso ciate d with the se quenc e: (a) F or every vertex v , | R + ( v ) | ≤ Q log n . (b) F or r = 2 , for every c or e ﬂ ipp able cycle C , X v ∈ C | R + ( v ) | ≤ Q log n . Remark 18. The pro o f o f Theorem 5 can b e e xtended to s how that w .h.p. for e very vertex v ∈ D , the subgraph induced by | R + ( v ) | has at most a s many ar cs as v ertices . 9 Remark 19. The case r = k = 2 follows from previously known work. F or c < c ∗ 2 , 2 = 1, it follows from the fact that w.h.p. every comp onent of G n,p = c/n has size O (log n ) b elow the giant comp onent thr eshold c ∗ 2 , 2 . F or c > c ∗ k,r , it follows from Lemma 5(b) of [24]. O ur pro of will w or k for k = r = 2, but it is conv enient to assume ( k , r ) 6 = (2 , 2 ). Remark 20. W e think that the conclusion o f Theorem 5 does not ho ld at c = c ∗ k,r + o (1). This is known to be true for the ca se k = r = 2. Indeed, whe n c = 1 − λ , for λ = n − 1 / 3+ ǫ , ǫ > 0, w.h.p. the size o f the lar gest comp onent is Θ( λ − 2 log n ) and no comp onent has mor e than o ne cycle [1 6]. A simple ﬁrst mo men t analysis yields that w.h.p. ther e is no cycle o f length greater than lo g n/ λ . F urthermo re, w.h.p. no vertex has degree greater than log n . It follows that the larg est co mpo nent m ust contain an induced subtree, none of whose vertices ar e in the 2- core, whic h has size Θ( λ − 2 log n log 2 n/λ ) = Θ(1 / ( λ log n )). It is easy to see that such a subtree will con tain v ertices with depth Θ(1 / ( λ log n )), whic h can b e as large as n α for a ny α < 1 / 3. The pro of o f Theor em 5 o ccupies Sections 7 and 8, after we set o ut some bas ic facts ab out co res in Section 5 and some bas ic calcula tions in Section 6. But ﬁrst, we show that it y ields Theo rems 3 and 4 : Pr o of of The or em 4. This follo ws immediately from Theorem 5 because the depth of v is at most | R + ( v ) | . W e a re now ready to give our algor ithm for trav eling b etw een any tw o ass ignments in the sa me cluster while c hanging O (log n ) v aria bles at a time. Pr o of of The or em 3. Giv en a n arbitrar y sy stem of linear equations consider a terminal 2 -stripping sequence v 1 , . . . , v t of its ass o ciated hypergr aph and let D b e the digraph formed fr om the sequence. F or ea ch core ﬂippable cycle , C , we choos e an arbitra ry vertex v C ∈ C . Let B be the s et consisting of each vertex v C and every non-2-cor e vertex with indegr ee zero in D . Consider an y 2-co re solution σ . Consider the system of equations formed from our sy stem by ﬁxing the v alue o f every 2-co re v aria ble that do es not b elong to a cor e ﬂippable cycle to its v alue in σ ; we c all such vertices ﬁxe d vertic es . Recall fro m Deﬁnition 4 that the edges of a ﬂippa ble cycle c ontain vertices that are not considered to b e vertices of the ﬂippable cycle; such vertices will b e ﬁxed. Note that the solutions of this system form a cluster, and that every c luster ca n b e formed in this wa y from some σ . W e will pe rform Ga ussian elimination o n this sy stem in a manner such that B will b e the se t of free v ar iables that we obtain. Importantly , this set of free v ariables do es not dep end on σ , i.e., it will b e the same for every cluster. F or each v ∈ B and fo r each ﬁxed vertex v , set χ ( v ) = { v } . F or each core ﬂippable cyc le C , we pro ce ss all of the edges (i.e., equations) joining consecutive vertices of C exce pt for one of the edge s cont aining v C . F or each v ertex v ∈ C , w e obtain the eq uation v = v C + z v where z v is a constant (0 or 1) dep ending only on the assignment to the ﬁxed vertices in the edges of C ; we set χ ( v ) = { v C } . By Lemma 35, the cor e ﬂippable cycles ar e vertex-disjoint, a nd so the eq uations corr esp onding to tw o co re ﬂippable cycles can overlap only on ﬁxed v a riables. Thus, we can carry this out for ea ch core ﬂippable cycle C independently . Next, we pro cess the edges not in the 2- core, in reverse remov al order , i.e ., E t , . . . , E 1 . Note that, since r = 2, ea ch E i contains at most o ne edge. When pro ce ssing E i , we set χ ( v i ) to the symmetric diﬀerence o f the sets χ ( u ), ov er all u ∈ E i other than v i . That is, a v ariable z is in χ ( v i ) iﬀ z ∈ χ ( u ) for an o dd n umber of v ar iables u ∈ E i other tha n v i . Since E i is the equa tion v i = P u ∈ E i ; u 6 = v u , this is equiv a lent (b y induction) to v i = P w ∈ χ ( v i ) w + z v i , where z v i is the sum of z u ov er all vertices u ∈ χ ( v i ) that b elo ng to core ﬂippable cycles. W e now note that every non-2-cor e v ertex v i / ∈ B has indegree a t least 1 in D and s o | E i | = 1 and th us χ ( v i ) is deﬁned. F or each vertex u 6 = v i in E i , either u ∈ B , or u is ﬁxed, or u = v j for some j > i , o r u is in a core ﬂippable cycle. The refore, b y induction, χ ( v i ) co ntains only vertices that are in B or ar e ﬁxed. Finally , note that po ssibly χ ( v i ) = ∅ ; in that case, v i = P w ∈ χ ( v i ) w + z v i = z v i in every solution. (It is not hard to ada pt the pro of of Theo rem 5 to show that w.h.p. for every i , χ ( v i ) 6 = ∅ . But that is not required for the purp oses of this pap er.) A t this p oint, all non-ﬁx ed v ertices are either in B or hav e bee n express ed as the sum of vertices in B and ﬁxed v ertices. Ther efore, the vertices in B are the free v ariables for the sys tem obtained by ﬁxing the v alues of the ﬁxed vertices to σ . Thus, there a re ex actly 2 | B | solutions to that s ystem, one for each assig nmen t to 10 B . W e ca n mov e betw een an y tw o such so lutions b y changing the assig nment s to the vertices of B , one at a time. Each time we c hange the v a lue of a non-2-co re v ertex v ∈ B , in order to get to a nother solution, we only need to change a subs et o f R + ( v ) in the dig raph D , b ecause o nly vertices u ∈ R + ( v ) can hav e v ∈ χ ( u ). Similarly , each time we change the v alue of some v C ∈ B , we only need to change a subset o f ∪ v ∈ C R + ( v ). Thu s, by Theo rem 5, we can mov e betw een a n y tw o such s olutions changing at most Q log n v a riables at a time. This implies Theorem 3, since each cluster is such a so lution set. W e close this s ection by sho wing how the preceding pro of extends to deter mine all of the froz en v ariables. A v a riable is sa id to b e fr ozen in a cluster, if it takes the same v alue in all assig nmen ts of the cluster . In general rando m CSP s it is hypothesized that the set of fro zen v aria bles can diﬀer from cluster to cluster. In random k -XOR-SA T, though, the s et of froz en v ar iables depends only on the underlying h yp erg raph, i.e., is the s ame for all c lusters. Theorem 6. In every cluster, the fr ozen variables c onsist of the 2-c or e vertic es n ot in c or e ﬂipp able cycles, and the non- 2-c or e variables v for which χ ( v ) ∩ B = ∅ . Pr o of. This follows immediately fro m the fact that B is the set o f free v a riables in a system of linear equations whose s olution se t is the cluster. 5 Random h yp ergraphs and their c ores W e will use the conﬁguratio n mo del of Bollo b´ as [6] to g enerate a rando m k -unifor m hyperg raph H with a given degree sequence. Supp os e we are given t he degree d ( v ) for each v ertex v ; th us P d ( v ) = k E wher e E is the num b er of hyperedges . W e take d ( v ) c opies of each v , and we take a uniformly r andom partition of these k E vertex-copies in to E sets of size k . This naturally yields a k -uniform hyper graph, b y mapping each k -set to a hyp eredge on the vertices whose copies are in the k -set. Note that the hypergra ph ma y contain lo ops (t wo copies of the sa me vertex in one hyperedge) and multiple edges (tw o ident ical hyper edges). It is well known that the pro bability that this par tition yields a simple hyper graph (i.e., one with no lo ops or multip le edges) is b ounded b elow b y a consta nt for degree sequences 1 satisfying certain conditio ns. Sp eciﬁcally: Deﬁnition 21. Say that a de gr e e se quenc e S is nice if E = Θ( n ) , P v d ( v ) 2 = O ( n ) and d ( v ) = o ( n 1 / 24 ) for al l v . Every degree sequence we will consider will corr esp ond to some s ubgraph of H k ( n, p ) with a linea r exp ected num b er o f edges. Since, as is well known, the degr ee sequence o f such random hyper graphs is nic e w.h.p., all the deg ree sequences we will consider will be nice. With this in mind, we will make heavy use of the following standar d prop os ition (see eg. [8]) and co rollar y , as working in the conﬁgur ation mo del is techn ically muc h ea sier than working with uniformly rando m hyperg raphs with a g iven degr ee seq uence. Prop ositio n 22. If S is a n ic e de gr e e se qu en c e, then ther e ex ists ǫ > 0 s u ch that the pr ob ability that a r andom hyp er gr aph with de gr e e s e quenc e S dr awn fr om the c onﬁgur ation mo del is simple is at le ast ǫ . This immediately yields: Corollary 23. If S is a nic e de gr e e se quenc e then: (a) If pr op erty Q holds w.h.p. for k -u niform hyp er gr aphs with de gr e e se qu enc e S dr awn fr om the c onﬁgu- r ation mo del, then Q holds w.h.p. for uniformly ra ndom simple hyp er gr aphs with de gr e e se quenc e S . (b) F or any r andom variable X , if E ( X ) = O (1) for k -uniform hyp er gr aphs with de gre e se quenc e S dr awn fr om the c onﬁgur ation mo del, then E ( X ) = O (1) for uniformly r andom simple hyp er gr aphs with de gr e e se qu en c e S . 1 Clearly , we are ref erring to a sequence of degree sequenc es S n so that asympto tic state ments are meaningful. W e suppress this p oint though, througho ut, to streamline exposition. 11 The following lemma will be v ery useful. Its exp onential term is not tight, but will suﬃce for our purp oses. Lemma 24 . Consider a r andom k -u niform hyp er gr aph dr awn fr om the c onﬁgur ation mo del with E e dges, i.e., with t otal de gr e e k E . F or e ach i = 2 , . . . , k , s p e cify ℓ i sets of i vertex-c opies, and set L = P k i =2 ℓ i . The pr ob ability t hat e ach of these sets app e ars in some hyp er e dge, and no two app e ar in the same hyp er e dge is less than exp  k L 2 E − L  k Y i =2  ( k − 1)( k − 2) · · · ( k − i + 1) ( k E ) i − 1  ℓ i . Pr o of. W e c ho ose the partition of t he vertex-copies by proc essing t he sp eciﬁed sets one-at-a -time. T o pro cess one o f the ℓ i sets o f size i , we ﬁrst cho ose one s et member γ ar bitrarily and then randomly se lect the remaining k − 1 v ertex -copies of the part containing γ . Every time w e do this there are at least k E − k L yet unselected vertex-copies. Th us, the pro bability w e c hose all o ther i − 1 mem b ers of the sp eciﬁed set is at mo st ( k − 1)( k − 2) · · · ( k − i + 1) ( k E − k L ) i − 1 < ( k − 1)( k − 2) · · · ( k − i + 1) ( k E ) i − 1 ×  E E − L  i − 1 < ( k − 1)( k − 2) · · · ( k − i + 1) ( k E ) i − 1 × e kL/ ( E − L ) , since i ≤ k . So the probability that ea ch o f the L tuples is chosen to b e in a hyper edge is less than k Y i =2  ( k − 1)( k − 2) · · · ( k − i + 1) ( k E ) i − 1  ℓ i × e ( kL/ ( E − L )) ℓ i = e kL 2 / ( E − L ) × k Y i =2  ( k − 1)( k − 2) · · · ( k − i + 1) ( k E ) i − 1  ℓ i . 5.1 Cores Recall fro m Section 4 tha t Theo rem 5 is alr eady known for k = r = 2. So we will assume that k + r > 4. It is well known that the r - core of a ra ndom k -uniform hyper graph is unifor mly r andom conditional on its degree sequence. See [25] for the ca se k = 2, and [21] for the nearly identical pr o of for g eneral k . In fact, the s ame is true of the gra ph remaining after any num be r of iteratio ns of the par allel stripping pro cess. Let H = H k ( n, p ) b e a random k -uniform hypergr aph and let H = H 0 , H 1 , . . . be the sequence o f hypergra phs pro duced by the parallel r -str ipping pro cess. It is well known how (see e.g., [2 1]) to s how the following prop ositions . Prop ositio n 25. (a) F or every i ≥ 0 , H i is u niformly r andom with r esp e ct to its de gr e e se quenc e. (b) Ther e exist functions ρ 0 , ρ 1 , . . . such t hat for a ny ﬁx e d inte ger i , w.h .p. H i c ontains ρ j ( i ) n + o ( n ) vertic es of de gr e e j and 1 k ( P j ≥ 1 j ρ j ( i )) n + o ( n ) e dges. Remark 26. The functions ρ j ( i ) hav e explicit recurs ive express ions, which w e g ive in Sectio n 8. An approximation is sta ted in Pro p osition 31 b elow. Prop ositio n 25 allows us to use the conﬁguration mo del to study H i . W e will b egin b y showing that we can unifor mly approximate the total degree of H i . Lemma 27. F or every ﬁxe d inte ger i ≥ 0 , X v ∈ H i deg H i ( v ) =   X j ≥ 1 j ρ j ( i )   n + o ( n ) . 12 Pr o of. Propo sition 2 5 implies that P j ≥ 1 j ρ j ( i ) is conv erg ent , else w.h.p. H i , a nd hence H , would have a sup erlinear n umber of edges. Consider any ﬁxed J . By P rop osition 2 5, w.h.p. P v :d eg H i ( v ) ≤ J deg H i ( v ) = P J j =1 j ρ j ( i ) n + o ( n ). F or any θ > 0 , the conv erg ence of P j ≥ 1 j ρ j ( i ) implies that we can choose J = J ( θ ) s uﬃcien tly lar ge that P j >J j ρ j ( i ) < θ / 2. Since H i ⊆ H 0 = H , we hav e P v :d eg H i ( v ) > J deg H i ( v ) ≤ P v :d eg H ( v ) > J deg H ( v ). The fact that the latter sum is less than θ n/ 2 for J suﬃcien tly la rge is well known and follows fro m the facts that (i) for ea ch constant ℓ , the num ber o f vertices of de gree ℓ in H is w.h.p. λ ℓ n + o ( n ) for a particula r λ ℓ = λ ℓ ( c ) and (ii) the num b er of hyperedge s in H is highly concentrated ar ound 1 k P ℓ ≥ 1 ℓλ ℓ n . Thu s,    P v ∈ H i deg H i ( v ) −  P j ≥ 1 j ρ j ( i )  n    < θ n for every θ > 0, which establishes the lemma. The following similar b ound will also b e useful: Lemma 28. F or every c onst ant d and ﬁxe d inte ger i > 0 : X v :d eg H i ( v ) ≥ d deg H i ( v )! (deg H i ( v ) − d )! =   X j ≥ d j ! ( j − d )! ρ j ( i )   n + o ( n ) . Pr o of. The pro of is almost identical to that of Lemma 27 but exploits the concentration o f the num b er of d -stars in H , ra ther than of the num b er of hyper edges. (A d -star is a s et of d h yp eredg es which contain a common vertex.) The concentration of the num b er of d -stars in H is easily established, e.g., b y the Sec ond Moment Method or T alag rand’s Inequality . (Indeed, Lemma 27 and its pro o f are s pec ial cas es of this lemma and its pro of for d = 1.) F or any ﬁxed integers k , r a nd rea l num b er λ > 0, we wr ite Ψ r ( λ ) = e − λ X i ≥ r − 1 λ i /i ! and f k,r ( λ ) = f ( λ ) = ( k − 1)! λ Ψ r ( λ ) k − 1 . Recall that for k + r > 4, the thresho ld for the a ppea rance of a n r -co re in a ra ndom k -uniform hypergra ph H k ( n, p ) with p = c/n k − 1 is c ∗ k,r = min λ> 0 f k,r ( λ ) . W e will see that f ′ has a unique ro ot a nd, thus, fo r c > c ∗ k,r the e quation f ( λ ) = c has t wo solutions. Deﬁnition 29. F or c > c ∗ k,r , let µ = µ ( c ) denote the lar ger of the two solutions of f ( λ ) = c . The following tw o prop ositions ar e standard; see e.g ., [2 1] for pro ofs. Prop ositio n 30. F or every ﬁ xe d j ≥ r , w.h.p. the r - c or e c ontains ( e − µ µ j /j !) n + o ( n ) vertic es of de gr e e j . F u rthermor e, w.h.p. the r -c or e c ontains ( µ/k )Ψ r ( µ ) n + o ( n ) e dges. Prop ositio n 31. F or every c 6 = c ∗ k,r and θ > 0 , ther e exists B = B ( θ ) such that w.h.p. (a) H B c ontains fewer t han θ n vertic es n ot in the r -c or e; (b) F or e ach j ≥ r , | ρ j ( B ) − e − µ µ j /j ! | < θ . The following lemma will b e critica l for our a nalysis. Lemma 32. F or every c > c ∗ k,r , t her e exists ζ = ζ ( k , r, c ) > 0 such that ( k − 1) µ r − 1 ( r − 2)! < (1 − ζ ) X i ≥ r − 1 µ i i ! , (2) wher e µ is t he lar ger of t he two r o ots of t he e quation f k,r ( λ ) = c . 13 Pr o of. f ′ ( λ ) = 0 ⇐ ⇒ Ψ r ( λ ) = λ ( k − 1)Ψ r ( λ ) k − 2 Ψ ′ r ( λ ) ⇐ ⇒ X i ≥ r − 1 λ i i ! = ( k − 1) λ r − 1 ( r − 2)! . (3) Equation (3) yields c ∗ k,r = f ( λ ∗ ) for some λ ∗ satisfying the last e quation in (3). F or c > c ∗ k,r , since µ = µ ( c ) is the larger of the tw o r o ots of f ( λ ) = c , it follows that µ > λ ∗ . The lemma now follows by no ting that the RHS o f (2) divided by the LHS is pro po rtional to P i ≥ r − 1 µ i − r +1 i ! , whic h is clear ly increas ing with µ . 6 Preliminaries to the pro of of Theorem 5 Recall that w e assume k + r > 4 and let H = H k ( n, p ) b e a rando m k -uniform hyper graph with p = c/n k − 1 . Let H = H 0 , H 1 , . . . b e the s equence o f hyper graphs pro duced by the parallel r -stripping pro cess. As we said ab ov e, we will choose a suﬃciently la rge constant B , strip down to H B , a nd then fo cus on R + ( u ) ∩ H B , making use o f the fact that H B is v ery close to the 2 -core (by Prop ositio n 31). The following will b e used to b ound the num b er of vertices that are r emov ed from R + ( u ) when stripping do wn to H B . F or int eger s ≥ 0, we use N s ( v ) to denote the s -th neighborho o d of v , i.e., the set of vertices within distance s from v . F or any set of vertices A , N s ( A ) = S v ∈ A N s ( v ). W e consider a single v ertex to b e a c onnected set. A s traightforw ar d induction yields the following. Prop ositio n 33. F or any inte ger i and vertex u ∈ H i , R + ( u ) ⊆ N i ( R + ( u ) ∩ H i ) . Lemma 34. F or any c, s ≥ 0 , ther e exists Γ = Γ( c, s ) such that in a r andom gr aph G ( n, p ) with p = c/n , w.h.p. for every c onne cte d s u bset A of vertic es | N s ( A ) | ≤ Γ( | A | + log n ) . Pr o of. W e prov e this for the cas e s = 1, i.e., that there is a constant γ > 1 such that w.h.p. every c onnected subset of vertices A satisﬁe s | N ( A ) | ≤ γ ( | A | + log n ). By itera ting, we obta in that for every s ≥ 1, every connected subs et o f vertices A satisﬁes | N s ( A ) | ≤ f s ( | A | ) where f 1 ( x ) = γ ( x + lo g n ) f i +1 ( x ) = γ ( f i ( x ) + log n ) , for i ≥ 1 . A s imple induction yields f i ( x ) ≤ γ i ( x + i lo g n ) and that yields the lemma with Γ = sγ s . Given any set A of size a , the pr obability tha t A is connected is at most the exp ected num b er of spanning trees of A which is a a − 2 ( c/n ) a − 1 . After conditioning that A is co nnected, the num b er of neighbo rs o utside of A is distributed a s Bin( a ( n − a ) , c/n ). The proba bilit y that this exceeds z is at most  a ( n − a ) z   c n  z <  eca z  z < 2 − z , for z > 2 eca . F or any γ > 2, if | N ( A ) | > γ ( | A | + log n ), then we must hav e | N ( A ) \ A | > 1 2 γ ( | A | + log n ). T aking γ > 4 ec , the e xpe cted num b er of co nnected sets A satisfying this last inequality is at most  n a  a a − 2  c n  a − 1 2 − 1 2 γ ( a +l og n ) < en a 2 ( ec ) a − 1 2 − 1 2 γ ( a +l og n ) < en a 2  ec 2 γ / 2  a − 1 2 − 1 2 γ log n = n − Θ( γ ) , for γ suﬃciently lar ge. Multiplying b y the n choices for a yields the lemma. Lemma 35. Fix k ≥ 3 and let H = H k ( n, p ) b e a r andom k -u niform hyp er gr aph with p = c/n k − 1 , wher e c > c ∗ k, 2 . (a) The ex p e cte d numb er of vertic es in c or e ﬂipp able cycles of H is O (1) . (b) W .h.p. no vertex lies in t wo c or e ﬂipp able cycles. 14 Pr o of. Let D b e the degree sequence of the 2-core o f H . By Corolla ry 23, w e c an work in the conﬁguratio n mo del. Reca lling Deﬁnition 29, Prop osition 3 0 a nd Lemma 32, w.h.p. (i) D has tota l degr ee γ n + o ( n ), where γ = µ Ψ r ( µ ), (ii) D has λ 2 n + o ( n ) vertices of degree 2, wher e λ 2 = e − µ µ 2 / 2, (iii) there exists ζ > 0 such that 2 ( k − 1) λ 2 < (1 − ζ ) γ . W e ﬁrst b ound the exp ected num be r of co re ﬂippable cycles of siz e a . Let Λ = γ n + o ( n ) b e the total nu mber of v ertex co pies, a nd let L = λ 2 n + o ( n ) b e the num b er of copie s of degree 2 vertices. There are  L a  choices for the co nnecting vertices, ( a − 1)! 2 wa ys to order them into a cycle, and 2 a wa ys to align their v ertex -copies. This y ields a pa irs { y 1 , z 1 } , . . . , { y a , z a } of vertex copies, each of whic h m ust land in a hyper edge. W e pro cess these pa irs o ne-at-a-time, halting if w e ev er ﬁnd that the pair do es not land in a hyperedge. T o pro cess pair i , we a sk only whether z i lands in the same hyper edge as y i ; if it do es we do not exp ose the other vertex-co pies in that hyperedge. Thus, prior to pro cessing pair i , we ha ve expo sed ex actly 2 i − 2 vertex-copies, all o f degree 2. There are k − 1 other co pies app earing in the sa me hyp eredge as y i . Each of the Λ − (2 i − 1) unexp os ed copies (not including y i ) is equally lik ely to b e one of tho se copies (and, for k ≥ 3, the exp osed c opies also ha ve p ositive probability). So the probability that z i is one of them is at most ( k − 1) / (Λ − 2 i + 1). So the exp ected num ber of core ﬂippable cycles of length a is at most:  L a  ( a − 1)! 2 2 a a Y i =1 k − 1 Λ − 2 i + 1 < 1 2 a a Y i =1 2( k − 1)( L − i + 1) Λ − 2 i + 1 . By co ndition (iii) ab ove, 2( k − 1) L/ (Λ − 1) < 1 − 1 2 ζ , and so 2( k − 1)( L − i + 1) / (Λ − 2 i + 1) < 1 − 1 2 ζ for each i , since L ≤ 1 2 (Λ − 1). So the expected num b er is at most 1 2 a (1 − 1 2 ζ ) a , and so the exp ected tota l nu mber o f vertices on co re ﬂippable cycles is at most 1 2 P a ≥ 1 (1 − 1 2 ζ ) a = O (1). This esta blishes par t (a). W e now prove part (b) b y using a ﬁ rs t momen t calculation. W e start b y sho wing that if t wo cor e ﬂipp able cycles hav e a common v ertex, then their union must con tain a simple structure: a ﬂippable cy cle plus a path. The pr o of will then follow since w.h.p. a ny such structure in a spa rse r andom hypergr aph must b e larger than what is pe rmitted by par t (a ). Intuitiv ely , this is stra ightforw ard, but the details are tedious. Recall that for every ﬂippable cycle C of a hyper graph H , every vertex of C ha s deg ree tw o in H and every edge of H contains either zero or exactly t wo vertices of C . L et S = x 1 , x 2 , . . . , x a be a core ﬂippable cycle and let S ′ 6 = S b e any other cor e ﬂippable cy cle that cont ains x 1 , where w ithout loss o f generality we assume | S ′ | ≤ | S | . Let e 1 be the e dge incident to S that co n tains x 1 , x 2 and let e a be the edge incident to S that contains x 1 , x a . Since e 1 , e a are the only tw o edges incident to x 1 , they must b e among the edges incident to S ′ . Let y 2 6 = x 1 be the other vertex of S ′ in e 1 . If y 2 = x 2 , let e 2 be the edge co nt aining x 2 , x 3 and let y 3 6 = y 2 be the other vertex of S ′ in e 2 . If y 3 = x 3 , let e 3 be the edge containing x 3 , etc. un til we ﬁrst reach an edge e j containing x j , x j +1 such that y j +1 6 = x j +1 . Let e ′ j +1 be the edge containing y j +1 , y j +2 . Now, let e ′ j +2 be the e dge containing y j +2 , y j +3 , let e ′ j +3 be the e dge containing y j +3 , y j +4 etc., until we ﬁrst reach an edge e ′ j +1+ q containing a vertex lying in a h yp eredg e incident to S (this must o ccur for some q ≥ 0 since e a is incident to S ′ ). Thus, we ha ve a path y j +1 , . . . , y j +1+ q +1 where y j +1 and y j +1+ q +1 lie in hyperedges incident to S , while all o ther q vertices y j +2 , . . . , y j +1+ q do not lie in hypere dges incident to S . In the following, to lighten notation, we let j + 1 = ℓ and j + 1 + q + 1 = ℓ ∗ , i.e., y ℓ and y ℓ ∗ are the v ertices in the path lying in hyperedg es incident to S . Now we b ound the exp ected num ber of o ccur ences of such a cycle S of s ize a plus a path y ℓ , ..., y ℓ ∗ , in H = H k ( n, p ). Since | S | = a there ar e at mo st a 2 choices fo r the v alue of ℓ and the v alue of z such that x z − 1 , x z , y ℓ share a h yp eredge. Since | S ′ | ≤ | S | there ar e at mo st a choices for the v alue of ℓ ∗ . Setting b = ℓ ∗ − ℓ + 1 to b e the num ber of v ertices in the p ath, the n um b er of c hoices for x 1 , ..., x a , y ℓ , ..., y ℓ ∗ is at mos t n a + b . The pro bability that x ℓ − 1 , x ℓ , y ℓ share a h yp eredg e in H is at most  n k − 3  c n k − 1 , and the sa me is true for x z − 1 , x z , y ℓ ∗ . F or each i 6 = ℓ, z , the probability that x i − 1 , x i share a hyperedge is at most  n k − 2  c n k − 1 . F or each i = ℓ , ..., ℓ ∗ − 1, the probability that y i , y i +1 share a hyperedge is at most  n k − 2  c n k − 1 . So the exp ected 15 nu mber of suc h s ubgraphs is at most: a 3 n a + b  n k − 3  c n k − 1  2  n k − 2  c n k − 1  a + b − 3 < a 3 c a + b − 1 n = o (1) , for a + b ≤ 1 2 log n . Th us w.h.p. there is no such subgr aph with a + b ≤ 1 2 log n . Since | S | ≥ | S ′ | , if a + b > 1 2 log n then | S | = a > 1 4 log n . But part (a) a nd Markov’s Inequality imply that w.h.p. there is no core ﬂippable cycle of s ize at least 1 4 log n . This prov es (b). Remark: By car rying out the ﬁrst moment calculatio n mo re carefully , as in pa rt (a), o ne obtains that the s um ov er all a, b of the ex pec ted num b er of o ccurr ences of a core ﬂippa ble cycle S plus a path y ℓ , ..., y ℓ ∗ as des crib ed is, in fact, O ( n − 1 ). 7 Pro of of Theorem 5 ab o v e the r -core threshold Recall that we ca n assume k + r > 4 . W e let H = H k ( n, p ) b e a rando m k -unifo rm hypergr aph with p = c/n k − 1 . Let H = H 0 , H 1 , . . . be the sequence of hyper graphs pr o duced by the parallel r -stripping pro cess. W e will cho ose a terminal r - stripping sequence that is co nsistent with the para llel pro cess; i.e ., in our stripping sequence: for every i < j , the vertices deleted in round i of the parallel pr o cess come b efor e the vertices deleted in round j of the parallel pro ce ss. Let D b e the digr aph asso cia ted with this ter minal r -stripping sequence and recall tha t R + ( u ) denotes the s et o f vertices rea chable fr om a vertex u in D . 7.1 Bound on t he length of stripping sequences Our ma in challenge is to pr ov e the following lemma. The idea is that we will take B large enoug h so that by stripping down to H B , Prop osition 31 gives us control o f the deg ree s equence that r emains, and Lemma 3 2 allows us to prove that a certain br anching pro ces s inv olving lo ng paths in a gra ph c onstructed from H B dies out. Lemma 36. F or every c > c ∗ k,r ther e exists B = B ( c, k , r ) and Q = Q ( c, k , r ) such that w.h.p. for every vertex u , | R + ( u ) ∩ H B | ≤ Q log n . Pr o of of The or em 5(a). Co nsider an y v ertex u . If u / ∈ H B , then b y Prop osition 33, R + ( u ) ⊆ N B ( u ) in whic h case Lemma 34 immediately implies that | R + ( u ) | < Γ(1 + lo g n ) for some constant Γ = Γ( c, B ). If u ∈ H B , then R + ( u ) ⊆ N B ( R + ( u ) ∩ H B ), by Pr op osition 33. Since, by Lemma 36, | R + ( u ) ∩ H B | ≤ Q lo g n , Lemma 34 now implies that | R + ( u ) | < Γ( Q log n + log n ) = Z log n for Z = Γ Q + 1 = Z ( c, B ) = Z ( c, k , r ). Deﬁnition 37. F or any i , we deﬁne D i to b e the sub digr aph of D induc e d by the vertic es in H i . Consider a particula r constant i . Let T + be a directed tree in D i with edg es direc ted awa y from a ro ot u that spans the v ertices of R + ( u ) ∩ H i ; e.g., T + could be a Breadth First Search or Depth First Sea rch tree from u . Thus, each vertex has indegree at most 1 in T + , implying: Prop ositio n 38. No two ar cs of T + wer e forme d during t he r emoval of the same hyp er e dge. Deﬁnition 39. A de letion tree r o oted at u is the un dir e cte d tre e, T , forme d by r emoving the dir e ctions fr om a tr e e T + r o ote d at u . T o prov e Lemma 36, w e will b ound the exp ected num b er of deletion trees T of size greater than Q log n . The follo wing technical lemma b ounds the dens it y of sma ll subgr aphs o f H k ( n, p ). It is of a standard ﬂa vour and has a sta ndard pro o f. Giv en a subset S of the vertices of H k ( n, p ), we let ℓ j ( S ) denote the nu mber of hyperedges that contain exactly j o f the vertices of S , and we let L ( S ) = P k j =2 ( j − 1) ℓ j . 16 Lemma 40. F or every c, ζ > 0 , t her e is θ > 0 , su ch that w.h.p. every S ⊆ H k ( n, p = c/n k − 1 ) with | S | ≤ θ n has L ( S ) < (1 + ζ ) | S | . Pr o of. Rather than w or king in the H k ( n, p ) mo del, it will b e con venien t to w or k in the H k ( n, m ) mo del, where exactly m = ( c/k !) n edges are selected unif or mly , indep endently a nd with r eplacement (note that m = p  n k  ). Standard arguments imply that high probability pr op erties in this mo del tra nsfer to the H k ( n, p ) mo del. Let Y a = Y a ( ζ ) denote the num ber of sets S with | S | = a a nd L ( S ) = (1 + ζ ) | S | . W e will b ound E ( Y a ) as follows. Deﬁne L a =    ( ℓ 2 , . . . , ℓ k ) : k X j =2 ( j − 1) ℓ j ≥ (1 + ζ ) a    . Cho ose a v ertices and some ( ℓ 2 , . . . , ℓ k ) ∈ L a , pick ℓ j edges for each j , and then multiply b y the probability that eac h edge choo ses (at least) the appropr iate num ber of vertices from S . This yields E ( Y a ) ≤  n a  X ( ℓ 2 ,...,ℓ k ) ∈L a k Y j =2  m ℓ j   k j   a n  j  ℓ j <  en a  a X ( ℓ 2 ,...,ℓ k ) ∈L a  a n  P k j =2 ( j − 1) ℓ j k Y j =2 ( J a ) ℓ j ℓ j ! , for some co nstant J = J ( c, k ) > 0 <  en a  a  a n  (1+ ζ ) a k Y j =2   X ℓ j ≥ 0 ( J a ) ℓ j ℓ j !   < e a  a n  ζ a e ( k − 1) J a =  ∆ a n  ζ a , for some co nstant ∆ = ∆( c, k , ζ ) > 0 . Cho osing θ = 1 2∆ , it is standard a nd stra ightf or ward to show E  P θ n a =1 Y a  = o (1). In order to carry out our ﬁrs t moment c alculation, we will b ound the diﬀerence b etw een the degr ees of the vertices of T and their degr ees in H i . Lemma 41. F or any δ > 0 , if i is suﬃciently lar ge in terms of δ then w.h.p.: F or every vertex u ∈ D i , if T is a deletion tr e e r o ote d at u , then deg H i ( v ) ≤ deg T ( v ) + r − 2 for al l but at m ost δ | T | + 3 vertic es v ∈ T . Pr o of. Deﬁne S to b e the hypergr aph with edge s et { e ∩ R + ( u ) : e ∈ H i , | e ∩ R + ( u ) | ≥ 2 } . In other words, for each hyperedg e e ∈ H i that co ntains at least tw o vertices o f R + ( u ), S contains the edg e o btained by removing all vertices o utside of R + ( u ) from e . Since V ( T ) = R + ( u ) ∩ H i , the r - stripping s equence that yields D c ontains an r -s tripping subse quence which removes from H i only v ertice s of T , such that all v ertices of T exc ept possibly u are remov ed. Consider v ∈ T , v 6 = u . At the po int that v is remov ed, it has degree at most r − 1 in what rema ins of H i . Every other hyperedge of H i containing v is remov ed before v , a nd th us must con tain another member of R + ( u ). At least one o f thos e r − 1 hyperedg es co n tains a nother vertex of R + ( u ), namely the parent of v in T . Therefore: deg H i ( v ) ≤ deg S ( v ) + r − 2 . F or 2 ≤ j ≤ k , let ℓ j denote the num b er of hyperedges with j vertices in S . All vertices of S , ex cept po ssibly u , are not in the r -core. So, b y Lemma 31(a), we know that for a n y θ > 0 w e can s elect i suﬃcien tly large in terms of θ so that | S | < θ n . If we pick θ suﬃciently small in terms of δ , then Lemma 40 implies that w.h.p., P k j =2 ( j − 1) ℓ j < (1 + δ / 2 ) | S | . So X v ∈ R + ( u ) deg S ( v ) = k X j =2 j ℓ j ≤ 2 k X j =2 ( j − 1) ℓ j < (2 + δ ) | S | = (2 + δ ) | T | . 17 Now the total T - degree of the vertices in R + ( u ) is 2 | T | − 2, since T is a tree with edges of s ize 2 that spans R + ( u ). So for i suﬃciently larg e in terms of δ , X v ∈ R + ( u ) deg S ( v ) − deg T ( v ) ≤ (2 + δ ) | T | − (2 | T | − 2) = δ | T | + 2 . So deg T ( v ) 6 = deg S ( v ) for at most δ | T | + 2 vertices v ∈ R + ( u ). Also, deg H i ( v ) ≤ deg S ( v ) + r − 2 for all but at most one v ∈ R + ( u ) (namely v = u ). This pr ov es the lemma. Pr o of of L emma 36. W e will ﬁx a constant δ > 0 that is suﬃciently s mall for v ar ious bounds to hold. W e also take B suﬃciently large for v a rious b ounds to hold, including Lemma 41 for i ≥ B . Le t X a = X a ( B ) b e the num be r of dele tion trees T in D B with a vertices. Our go al is to show that there exists some co nstant Q > 0 such that w.h.p. X a = 0 for a > Q log n , so in the fo llowing we may allow ourselves to assume that a is gr eater than some suﬃcien tly large constant. T o prove Lemma 36 we ﬁrst o bserve that, by Prop os ition 31, w e can assume H B is uniformly random conditional on its degree seq uence. Since Lemma 36 a sserts a pr op erty to hold with high pr obability , it suﬃces to establis h this pr op erty in the conﬁguratio n mo del for H B (b y Co rollary 23(a)). More ov er, r ecall that b y P rop osition 3 1(b), as B is increas ed w.h.p. the degr ee sequence of H B tends to that of the r -core. Let v 1 , . . . , v a be the vertices of T . W e ﬁr st sp ecify d i = deg T ( v i ) fo r ea ch i , noting that these de grees m ust sum to 2 a − 2. The nu mber of wa ys to arra nge these a vertices int o a tree with a spe ciﬁed degree sequence is ( a − 2)! / Q ( d i − 1)! and there are a choices for the ro ot, u , of the tree . So, the num b er of choices for this step is: a ( a − 2)! Q ( d i − 1)! . Next we choose the vertices of T . Then for each edge of T , we choos e a vertex-copy o f each of its endpo int s. T o do so, for each v i , we choose a copy o f v i for each of the d i edges in T incident with v i . If deg H B ( v i ) = j , then there are j ! / ( j − d i )! choices for the d i copies of v i . Since deg H B ( v i ) ≥ d i , the n umber of choices co rresp onding to v i is at most P w : deg H B ( w ) ≥ d i deg H B ( w )! / (deg H B ( w ) − d i )!. By Lemma 28, this nu mber is at most ( Y ( d i ) + 1 2 δ ) n wher e Y ( d ) = Y B ( d ) = X j ≥ d j ! ( j − d )! ρ j ( B ) . F urthermor e, if d i ≤ deg H B ( v ) ≤ d i + r − 2, then w e can use Y ′ ( d i ) r ather than Y ( d i ) wher e Y ′ ( d ) = Y ′ B ( d ) = d + r − 2 X j = d j ! ( j − d )! ρ j ( B ) . Using Y ′ ( d i ) instea d of Y ( d i ) will b e particularly useful when d i ≤ 2. By Lemma 4 1, for any δ > 0 we can take B = B ( δ ) > 0 suﬃciently lar ge, so that we must use Y ( d i ) for a t most δ a + 3 vertices v i . F or conv enience, we will assume a > 3 /δ so we can take δ a + 3 ≤ 2 δ a . W e will upp er b ound E ( X a ) b y using Y ( d i ) for every vertex v i with d i ≥ 3 and for exactly 2 δ a vertices of degree d ≤ 2. Le t t 1 , t 2 , t 3 denote the num ber of vertices v i for which d i = 1 , d i = 2 , d i ≥ 3, resp ectively . W e note that for suﬃcien tly lar ge d , Y ( d ) is decreasing and so there is a constant d ∗ such that for all d ≥ 3, Y ( d ) ≤ Y ( d ∗ ). So, if we w ere to use Y ′ ( d ) for every vertex o f degree d ≤ 2 then the ov erall contribution of the Y , Y ′ terms w ould be at most: [( Y ′ (1) + 1 2 δ ) n ] t 1 · [( Y ′ (2) + 1 2 δ ) n ] t 2 · [( Y ( d ∗ ) + 1 2 δ ) n ] t 3 . W e cor rect for the 2 δ a vertices of degre e d ≤ 2 for which we use Y ( d ). T o do so , we multiply by the  t 1 + t 2 2 δa  ≤  a 2 δa  choices for those vertices, and w e m ultiply b y Υ 2 δa where, for δ suﬃciently small, Υ = max  Y (1) + 1 2 δ Y ′ (1) + 1 2 δ , Y (2) + 1 2 δ Y ′ (2) + 1 2 δ  = O (1) . 18 This brings the ov erall contribution of the Y , Y ′ terms to at most:  a 2 δ a  Υ 2 δa [( Y (1) + 1 2 δ ) n ] t 1 [( Y (2) + 1 2 δ ) n ] t 2 [( Y ( d ∗ ) + 1 2 δ ) n ] t 3 . Having c hosen d 1 , . . . , d a and the vertices v 1 , . . . , v a , we divide by the n umber of rearra ngements o f those vertices; i.e. we multiply by 1 a ! . Finally , we multiply by the pr obability tha t ea ch of the a − 1 pairs o f vertex-copies corr esp onding to edges of T , lands in a h yp eredg e of the co nﬁguration. By Prop os ition 38, no tw o s uch pairs lie in the same hyperedge of H B . So, we can a pply Lemma 24 to the a − 1 sp eciﬁed pairs of vertex-copies and m ultiply by  k − 1 k E  a − 1 e ka 2 / ( E − a ) to get an ov erall b ound, where E is the num b er of edg es in H B . Recall that for c > c ∗ k,r , µ = µ ( c ) denotes the larger of the tw o s olutions o f f ( λ ) = c . By Prop osition 3 1 and Lemma 27 for a ny δ > 0, we can take B suﬃciently large so that       k E − µ X j ≥ r − 1 e − µ µ j j ! n       ≤ δ n . Our key Lemma 32 now yields that b y taking B suﬃciently la rge, we can hav e δ suﬃciently small in terms o f ζ that v ario us bo unds below hold, including  e − µ µ r ( r − 2)! + δ  k − 1 k E /n < 1 − ζ 2 . (4) By Lemma 31, for an y δ > 0, w e can tak e B suﬃciently large so that Y ′ (1) ≤ δ / 2 and Y ′ (2) ≤ e − µ µ r ( r − 2)! + δ/ 2 . So, Y ′ (1) + 1 2 δ, Y ′ (2) + 1 2 δ ar e b ounded ab ov e by δ and e − µ µ r ( r − 2)! + δ , resp ectively . W e let Ψ = 2 Y ( d ∗ ) > Y ( d ∗ ) + 1 2 δ , for δ s uﬃcien tly small. Putting all this together, a nd r ecalling that t 1 + t 2 + t 3 = a , yields E ( X a ) ≤  a 2 δ a  Υ 2 δa  k − 1 k E  a − 1 e ka 2 / ( E − a ) × X d 1 + ··· + d a =2 a − 2 ( δ n ) t 1  e − µ µ r ( r − 2)! + δ  n  t 2 (Ψ n ) t 3 a ( a − 2)! a ! Q a i =1 ( d i − 1)! (5) ≤ O ( n/a ) e ka 2 / ( E − a )  Υ 2 δ (2 δ ) 2 δ (1 − 2 δ ) 1 − 2 δ  a  k − 1 k E /n  a X d 1 + ··· + d a =2 a − 2 δ t 1  e − µ µ r ( r − 2)! + δ  t 2 Ψ t 3 . Note that in the last line, w e dr opp ed the Q a i =1 ( d i − 1)! term. W e ca n aﬀord to do so, since this is equal to 1 for d i = 1 or 2, which a re the most sensitive v alues. F or δ suﬃciently sma ll in terms o f ζ , Υ 2 δ (2 δ ) 2 δ (1 − 2 δ ) 1 − 2 δ < 1 + ζ 10 . 19 Since we a re dea ling with the deg ree sequence o f a tree, we hav e t 1 > t 3 . Since δ < 1, w e hav e √ δ t 1 < √ δ t 3 , yielding: E ( X a ) < O ( n/ a ) e ka 2 / ( E − a )  1 + ζ 10  a × X d 1 + ··· + d a =2 a − 2  √ δ k − 1 k E /n  t 1  e − µ µ r ( r − 2)! + δ  k − 1 k E /n  t 2  √ δ Ψ k − 1 k E /n  t 3 . Recalling that E /n = Ω(1) and Ψ = O (1), we choose δ s uﬃcien tly small in terms of ζ s o that √ δ k − 1 k E /n , √ δ Ψ k − 1 k E /n < ζ 100 . This and (4) y ield E ( X a ) ≤ O ( n/ a ) e ka 2 / ( E − a )  1 + ζ 10  a X d 1 + ··· + d a =2 a − 2  1 − ζ 2  t 2  ζ 100  a − t 2 . Now we ﬁx t 2 and count the num b er of choices for d 1 , . . . , d a . Ther e ar e  a t 2  choices for the v alue s of i with d i = 2. The rema ining a − t 2 degrees s um to 2 a − 2 − 2 t 2 . The n umber of choices for sequences of y non-neg ative int eger s that sum to z is  y + z − 1 y − 1  , so the num b er of choices for thes e degrees is b ounded by  2( a − t 2 ) − 3 a − t 2 − 1  < 2 2( a − t 2 ) − 3 < 4 a − t 2 . Thus, E ( X a ) ≤ O ( n/ a ) e ka 2 / ( E − a )  1 + ζ 10  a a X t 2 =0  a t 2  4 a − t 2  1 − ζ 2  t 2  ζ 100  a − t 2 = O ( n/a ) e ka 2 / ( E − a )  1 + ζ 10  a  1 − ζ 2 + ζ 25  a < O ( n/a ) e ka 2 / ( E − a )  1 − ζ 4  a < O ( n/a )  1 − ζ 16  a , (6) where the last inequa lit y holds for all a small enough that e ka/ ( E − a ) < 1 + ζ 4 . Th us, there are constants Q, ξ > 0 such that E ( P ξn a = Q log n X a ) = o (1) and, therefore, w.h.p. there a re no deletion tre es of size betw een Q lo g n and ξ n . Note now that Q , ξ dep end only on ζ , c, k , r and ζ dep ends only on c, k , r . Using Prop o sition 31(a), we c hose B large enough that w.h.p. H B contains f ewer than ξ n vertices outside of the r -cor e. Since a deletion tree c an hav e at mos t one vertex in the r -cor e, this implies that there a re no deletion trees o f s ize at least ξ n . Therefor e, w.h.p. there a re no deletion tr ees in H B of size greater than Q lo g n . Therefore, w.h.p. fo r all u ∈ D B , | R + ( u ) ∩ H B | ≤ Q log n .  7.2 Summing ov er a cor e ﬂippable cycle for r = 2 Recall that for T heorem 5(b), we hav e r = 2; i.e ., we c onsider 2-c ores for r andom k -unifor m hyperg raphs where k ≥ 3. Consider any core ﬂippable cycle C with vertices u 1 , . . . , u ℓ . In o ur directed g raph D , a dd edges fro m u j to u j +1 for each j (addition mo d ℓ ). Thus, R + ( u 1 ) = ∪ ℓ j =1 R + ( u j ). W e mo dify the arg umen ts from the pro of o f pa rt (a) for this setting. W e deﬁne T as in the previous section, this time ro oted at u 1 . 20 W e follow the pro of of Lemma 4 1. Since u 1 , . . . , u ℓ are the o nly 2- core vertices in S , w e still have | S | ≤ θ n . Since each u i has deg ree 2 in the 2- core, it is easy to see that deg H B ( u i ) = deg S ( u i ) + 1. The pro of o f Le mma 41 still holds , y ielding deg H B ( v ) ≤ deg T ( v ) + 1 , for all but a t most δ | T | + 3 vertices v ∈ T . (In fac t, this time w e actually g et δ | T | + 2, but tha t is inconsequential.) As in Section 7 .1, w e b ound the exp ected n umber o f such tr ees of s ize a ; u 1 is the ro ot and he nce pla ys the role of u from Sectio n 7.1. This time, T ha s the additiona l prope rty that there is an edge in D from a vertex of T (i.e. u ℓ ) to u 1 . T o acco un t for this additiona l prop er t y , we adjust (5) as follows: (i) multiply b y the num be r of choices of one o f the a − 1 o ther vertices to b e u ℓ ; (ii) cho ose vertex-copies for the extra edge from u ℓ to u 1 ; (iii) a djust the ter m  k − 1 kE  a − 1 e ka 2 / ( E − a ) which, by Lemma 24, b ounded the probability that the a − 1 pairs of vertex-copies c orresp onding to edges of T each la nded in a h yp eredge of the conﬁgura tion. F or (ii), we use Y ( d ( u j ) + 1) instead of Y ( d ( u j )) or Y ′ ( d ( u j )) for j = 1 , ℓ . F or j = 1 , ℓ , the adjustment for u j is an incr ease of a multiplicativ e factor o f a t mo st ( Y ( d ( u j ) + 1) + 1 2 δ ) / ( Y ′ (deg( u j )) + 1 2 δ ) < ( Y ( d ∗ ) + 1 2 δ ) / ( Y ′ (1) + 1 2 δ ) = O (1). So the ov erall eﬀect for (ii) is a multiplicativ e O (1). F or (iii), the hyper edge con taining u 1 , u ℓ is in the 2-core and so is dis tinct from the other a − 1 hyperedg es. This results in another multiplicative factor of k − 1 kE to account for that edge, when applying Lemma 2 4. The net result is to multip y E ( X a ) by O ( a/n ), and so the b ound on E ( X a ) in (6) becomes O (1)  1 − ζ 16  a . Summing ov er all a y ields that the expected n umber of core ﬂippable cycles C such that | S u ∈ C R + ( u ) ∩ H B | > ξ ( n ) is o (1) fo r a ny ξ ( n ) → ∞ , in pa rticular for ξ ( n ) = O (lo g n ). P rop osition 33 and Lemma 34 yield Theorem 5(b).  8 Pro of of Theorem 5 b elo w the r -core threshold Recall that Theorem 5 is alr eady k nown for r = k = 2 , so we will as sume r + k > 4. As in the ca se for c > c ∗ k,r , we will carry out a la rge but ﬁx ed num b er, I , of ro unds of the paralle l r -stripping pr o cess, e nding up with a hyper graph H I . B ecause we ar e below the r -core threshold, this will delete a ll but a very sma ll, alb eit line ar, n umber of vertices. Pr op osition 25 asserts that the r emaining h yp ergr aph is uniformly random conditional on its degree sequence. W e will determine this degree sequence and apply the tec hnique from [22] to show that the maxim um comp onent size in the remaining h yp erg raph has s ize O (log n ). Thus, for every v , w e must ha ve | R + ( v ) ∩ H I | = O (log n ). Pr op osition 33 and Lemma 34 then imply that | R + ( v ) | = O (log n ) as r equired. Let Po( µ ) denote a Poisso n v aria ble with mean µ . Recursively deﬁne the following quantities: φ 0 = 1 λ t = cφ k − 1 t / ( k − 1)! φ t = Pr[Po ( λ t − 1 ) ≥ r − 1] . W rite P ( µ, j ) = Pr[Po( µ ) = j ]. Lemma 42. F or any c onstants d, t , the numb er of vertic es of de gr e e d after t r oun ds of the p ar al lel r -stripping pr o c ess, w.h.p. is ρ t ( d ) n + o ( n ) , wher e ρ t ( d ) =      P ( λ t , d ) for d ≥ r , P ( λ t , d ) · Pr [Po( λ t − 1 − λ t ) ≥ r − d ] for d < r . Pr o of. W e consider a bra nching pro cess introduced in [25] and analyze it as in [21]. Co nsider an y h yp ergr aph H a nd any vertex v ∈ H . F or ea ch 0 ≤ i ≤ t + 1, let L i ( v ) b e the vertices of distance at mo st i from v (thus L 0 ( v ) = { v } ). F or any u ∈ L i ( v ) with 0 ≤ i ≤ t , a child e dge of u is an edge containing u and k − 1 members 21 of L i +1 ( v ); thus if the dista nce t + 1 neigh b ourho o d of v induces a h yp ertree, then all but at most o ne of the edges c ontaining u a re child edges of u . W e deﬁne the pro cess STRIP( v , t ) as follows: F or j from t down to 1 do Remov e all vertices in L j ( v ) with fewer than r − 1 child edges; Remov e all edges tha t co nt ain a re mov ed vertex. Let X t denote the n umber of c hild edges o f v that sur vive STRIP( v, t ), a nd let Y t denote the num b er o f c hild edges of v that sur vive STRIP ( v , t − 1) but not STRIP( v , t ). If the hyperg raph induced by the vertices in L t +1 ( v ) induces a hypertree, then we see that (A) F or d ≥ r : v s urvives the ﬁrst t rounds of the pa rallel r -stripping pr o cess, and has deg ree d in wha t remains iﬀ X t = d . (B) F or 1 ≤ d < r : v s urvives the ﬁrst t rounds o f the par allel r -stripping pro ces s, and has degree d in what rema ins iﬀ X t = d and Y t ≥ r − d . T o ana lyze STRIP( v , t ) o n H = H k ( n, p = c/n k − 1 ), we make use of the fact that w.h.p. the distance t + 1 neighbourho o d of v induces a hype rtree, and so both (A) and (B) hold. W e will ar gue by induction on t that the pro bability a pa rticular child u of v survives STRIP( v , t ) is φ t + o (1). Supp ose u ∈ L 1 ( v ) and w ∈ L 2 ( v ) is in a child edge of u . Note that w survives STRIP( v , t ) iﬀ w survives STRIP( u, t − 1) the pro bability of whic h, by induction o n t , is eas ily seen to b e φ t − 1 + o (1). It follows that the exp ected num b er of child edges o f u that survive STRIP( v , t ) is c n k − 1  n − O (1) k − 1  ( φ t − 1 + o (1)) k − 1 = λ t − 1 + o (1). Standar d arg ument s show that for any ﬁxed i the probability that the num b er of such edges is i is P ( λ t − 1 , i ) + o (1) (we ela bo rate more below o n similar ar guments for X t , Y t ). Therefore, the probability that u survives STRIP( v , t ) is φ t + o (1), thus completing the induction. By the same argumen t, E ( X t ) = λ t + o (1). Noting that Y t = X t − 1 − X t , this yields E ( Y t ) = λ t − 1 − λ t + o (1). Consider any child edge e of v in L t +1 ( v ). Whether e counts tow ards X t , Y t or neither is determined ent irely by the subtrees of L t +1 ( v ) ro oted a t the vertices of e o ther than v . In o ther words, X t , Y t are determined by the e dges containing v in H k ( n, p = c/ n k − 1 ), and so me lo cal infor mation a bo ut ea ch edge where the informatio n for a ny tw o edg es is w.h.p. disjoint. Also, no edge c ounts towards bo th X t and Y t . F ro m this, it is straightforward to show, using e.g., the Metho d of Mo ment s, (see [14]) that the jo in t distribution of X t , Y t is as ymptotic to indep endent Poisson v ar iables; sp eciﬁcally , for any ﬁxed integers x, y , Pr ( X t = x ∧ Y t = y ) is o (1) plus the pro bability that tw o indep endent Poisson v aria bles with means E ( X t ) , E ( Y t ) a re equal to x, y . (A) and (B) now yie ld that the probability tha t v survives the ﬁrst t rounds o f the par allel stripping pro cess and has degre e d in H t is ρ t ( d ) + o (1), and s o the expected nu mber of such v ertices is ρ t ( d ) n + o ( n ). The lemma now follows as in [21] from a str aightforw ard concentration argument, e.g., a se cond moment calculation. W e omit the details. The main result of [22] states: Consider a r andom graph on a ﬁxed degree se quence where Λ( d ) · n + o ( n ) vertices ha ve degree d , and where the deg ree sequence satisﬁes certain wel l-b ehave d co nditions. If X d ≥ 1 d (2 − d )Λ( d ) > 0 , (7) and then w.h.p. all connected comp onents hav e size O (log n ). A simple adaptation of the pro o f in [22] provides a gener alization to hypergra phs. Sp eciﬁca lly , for k > 2 it suﬃces to replace d (2 − d ) in (7) with f k ( d ) = d [1 − ( d − 1)( k − 1)] . Prop ositio n 2 5 allows us to mode l H t as a random hypergra ph o n degree sequence ρ 0 ( t ) , ρ 1 ( t ) , ... . Using Lemma 28, it is straightforw ar d to v erify that this deg ree sequence satisﬁes the w ell-b ehaved co nditions from [22], a nd so deduce that if X d ≥ 1 ρ t ( d ) f k ( d ) > 0 , (8) 22 then w.h.p. all compo nents of H t hav e size O (lo g n ). Since Pr[Po( λ ) ≥ r − 1] is a strictly increasing function of λ , the sequences φ t , λ t are strictly decreasing . If they do not tend to zer o, then there must b e a p ositive ﬁxed p oint to the recur sion deﬁning them, i.e., a po sitive solution to λ = c Pr[Po ( λ ) ≥ r − 1] k − 1 / ( k − 1)! . Rearra nging yields c = ( k − 1)! λ/ P r[Po ( λ ) ≥ r − 1] k − 1 . Recall now that c ∗ k,r was deﬁned in (1 ) a s the smalle st v alue of c for which there is such a solution. Since c < c ∗ k,r , we can co nclude that φ t , λ t → 0 as t → ∞ and we can develop the following asymptotics in t , using O t () a nd Θ t () to denote asymptotics a re with resp ect to t : λ t = c ( k − 1)! φ k − 1 t = c ( k − 1)! (Pr[Po ( λ t − 1 ) ≥ r − 1]) k − 1 = Θ t ( λ ( k − 1)( r − 1) t − 1 ) . (9) Let λ := λ t and θ := λ t − 1 − λ t . Since ( k − 1)( r − 1) ≥ 2 fo r k + r > 4, we see that (9) implies λ = o t ( θ ). W e a pply Lemma 4 2, noting that as θ → 0 , P r[Po ( θ ) ≥ r − d ] → P ( θ , r − d ). Therefor e, as t → ∞ , inequality (8) is equiv alent to (1 + o t (1)) r − 1 X d =1 P ( λ, d ) P ( θ , r − d ) f k ( d ) + ∞ X d = r P ( λ, d ) f k ( d ) > 0 . (10) Note that f k (1) = 1 and f k ( d ) ≤ 0 for d ≥ 2 . So the ﬁrst sum in (10) is a t least P ( λ, 1) P ( θ, r − 1) − r − 1 X d =2 P ( λ, d ) P ( θ , r − d ) | f k ( d ) | . (11) F or 1 ≤ d ≤ r − 1, we hav e f k ( d ) = O t (1), s o the ﬁrst ter m in (11) is Θ t ( λθ r − 1 ) while the sum in (11) is Θ t ( P r − 1 d =2 λ d θ r − d ) = Θ t ( λ 2 θ r − 2 ), since λ = o t ( θ ). Therefore, the ﬁrst s um in (10) is p ositive a nd of o rder Θ t ( λθ r − 1 ). A t the same time, since − f k ( d ) = k d 2 − d 2 − k d < kd 2 we get − ∞ X d = r P ( λ, d ) f k ( d ) ≤ k ∞ X d = r P ( λ, d ) d 2 = O t ( λ r ) , as λ → 0 . Thu s, the ﬁrs t sum in (10) is po sitive Θ t ( λθ r − 1 ), wherea s the seco nd sum is O t ( λ r ). Since λ = o t ( θ ) it follows that (8) holds for t suﬃciently large a nd, therefore , for I suﬃciently large, every comp onent of H I has s ize O (log n ). Theore m 5(b) now follows from Pr op osition 33 and Le mma 34.  9 Pro of of Theorem 2 Given a solutio n, recall that a set S of v aria bles is ﬂipp able if changing the assignment o f every v a riable in S results in ano ther so lution. Note that ﬂippable sets ca n b e characterized in terms of the underly ing hypergra ph. Prop ositio n 43. S is ﬂipp able iﬀ every hyp er e dge c ontains an even n umb er of memb ers of S . So we deﬁne: Deﬁnition 44. A ﬂippable set in a hyp er gr aph, H , is a nonempty set of vertic es, S , such t hat every e dge in H c ontains an even nu mb er of vertic es of S . 23 Recalling Deﬁnition 4, we s ee that a ﬂippable cycle is a ﬂippa ble set. A ﬂippable s et is minimal if it do es not con tain a ﬂippable prop er subset. Note that every ﬂippable se t co nt ains a minimal ﬂippable subset. Lemma 45. L et H b e a r andom k -un iform hyp er gr aph H k ( n, p ) , wher e p = c/n k − 1 . F or every c > c ∗ k, 2 ther e exists α > 0 such that w.h.p. every minimal ﬂ ipp able set in the hyp er gr aph induc e d by t he 2-c or e of H either is a c or e ﬂipp able cycle or has size at le ast αn . Lemma 45 follows immediately from Lemma 51 b elow, a nd yields Theorem 2 as follows: Pr o of of The or em 2. Consider a ny tw o solutions σ 1 , σ 2 in diﬀere n t clusters. Let S b e the v ar iables in the 2-core on whic h these so lutions disagree. Thus, S is a ﬂippable set in the hyperg raph induced by the 2-cor e. Remov e all co re ﬂippable cycles from S , and let S ′ be what remains (recall from Deﬁnition 4 tha t a ﬂippable cycle is a set of vertices). Lemma 35(b) implies that w.h.p. every 2-core hyp eredge cont ains an even n umber of vertices that lie in cor e ﬂippa ble cycles . Therefore S ′ m ust also b e a ﬂippable set in the hyperg raph induced by the 2 -core. By the deﬁnition o f c lusters, S ′ 6 = ∅ as o therwise σ 1 , σ 2 would b e cycle- equiv ale n t. Let S ′′ be a mimimal ﬂippable subset of S ′ . Since S ′′ contains no core ﬂippable cycles, Lemma 4 5 implies that w.h.p. | S ′′ | ≥ αn . Therefore | S | ≥ αn a nd so σ 1 , σ 2 diﬀer on at least αn v ariables. An y seq uence σ , σ ′ , . . . , τ wher e σ, τ a re in diﬀerent clusters must contain tw o consecutive solutions that are in diﬀerent clusters. As a rgued ab ov e, those tw o so lutions diﬀer on at lea st αn v ariables . It follows that if σ, τ are in diﬀer ent clus ters then σ, t are not αn -connected. If w e could s how (deterministically) that the hypergra ph induced b y any minimal ﬂippable set in a 2-core that is not a core ﬂippable cycle is suﬃciently dense, then Lemma 45 w ould follow by a r ather standar d argument. Unfortunately , there is no useful low er bo und on the density , mainly b ecaus e of the p ossibility o f very long 2- linked paths in S (deﬁned b elow). Instead, we follow an appro ach akin to that of [23], forming a gra ph Γ( S ) b y co n trac ting those long pa ths, and making use of the fact that Γ( S ) is dense (Lemma 50). The main diﬀerence from [2 3] is that here we need to w ork in the conﬁguratio n mo del. T o prov e Lemma 4 5, we ﬁrst r equire a few deﬁnitions. Note that these concern any hyperg raph, not just a 2- core of a random hyperg raph. A hyperedg e is simple if it is not a lo o p, i.e., if it do es not c ontain any v ertex mo re than once. Deﬁnition 46. L et H b e a k -uniform hyp er gr aph. A 2-linked path P of a set S ⊆ V ( H ) is a set of vertic es v 0 , . . . , v t ∈ S and simple hyp er e dges e 1 , . . . , e t , wher e t ≥ 1 , such t hat (i) v 0 , . . . , v t ar e al l distinct exc ept that when t ≥ 2 we al low v 0 = v t . (Note that if v 0 = v t then t hese vertic es actual ly form a cycle and so 2-linked path is somewhat of a misnomer.) (ii) Ea ch e i c ontains v i − 1 , v i and no other vertic es of S . (iii) v 1 , . . . , v t − 1 al l have de gr e e 2 in H ; i.e. they do not lie in any e dges outside of P . (iv) If v 0 = v t then deg H ( v 0 ) > 2 . If v 0 6 = v t then S is maximal w.r.t. (ii) and (iii); i.e. e ach of v 0 , v t either has de gr e e 6 = 2 in H , or lies in a hyp er e dge e / ∈ P with | e ∩ S | 6 = 2 . We c al l v 0 , v t the endp oints of the p ath and v 1 , . . . , v t − 1 its co nnecting vertices . Note that if v 0 = v t then b y (iv), deg H ( v 0 ) > 2 and hence v 0 , . . . , v t do not form a ﬂippable cycle. Figure 1: 2-linked pa ths with t = 3. On the left v 0 6 = v 3 , while on the right v 0 = v 3 . V ertices in S a re marked with a squar e. Deﬁnition 47. We say that S ⊆ V ( H ) is a linked set if (i) S do es not c ontain a ﬂipp able cycle as a subset, (ii) no hyp er e dge of H c ontains exactly one element of S and (iii) every hyp er e dge e of H with | e ∩ S | = 2 is in a 2-linke d p ath of S . 24 Prop ositio n 48. Supp ose S is a ﬂipp able set in a hyp er gr aph wher e al l hyp er e dges ar e simple, and S do es not c ontain a ﬂipp able cycle as a subset. Then S is a linke d set. Pr o of. By Pr op osition 43, we only need to chec k condition (iii) of Deﬁnition 47. Consider any hyper edge e with | e ∩ S | = 2. Since e is simple, either e itself for ms a 2- linked path in S , or it is easily seen that e can be extended into such a pa th, unless e lie s in a ﬂippable cycle. Remark: I t is easy to s ee that in any Uniquely E xtendible C SP , the set of dis agreeing v ar iables of any t wo s olutions m ust b e a ﬂippable set. Since P rop osition 48 was derived by only consider ing the underlying hypergra ph (and not the sp eciﬁc constraints), it applies to a n y UE CSP . Therefore, our Theorem 2 extends readily to every UE CSP since its pro of amounts to proving that for some constant α > 0, all linked sets are either ﬂippable cycles or con tain at least αn v ariables . Given a linked set, S , we consider the mixe d hyper graph (containing b oth hyperedges and no rmal edg es) Γ( S ) formed as follows: (a) The vertices of Γ( S ) are the endp oints of the 2-linked paths in S alo ng with all vertices of S that do not lie in any 2-linked paths. (b) There is a n edge in Γ( S ) b etw een the endp oints of each 2- linked path in S . That edge is a loo p if the t wo endpo int s are the s ame vertex, and so Γ ( S ) is no t nec essarily s imple. (c) F or every hype redge e of H with | e ∩ S | > 2 , e ∩ S is a hypere dge of Γ( S ). Thu s V (Γ( S )) ⊆ S , and since no h yper edge of C contains exactly one element of S , for every v ∈ V (Γ( S )) we hav e deg Γ( S ) ( v ) = deg H ( v ). Any v ertex of S that is no t in Γ( S ) is a co nnecting vertex of a 2- linked path in S . Prop ositio n 49. If S is a non-empty linke d set, then 1 ≤ | Γ( S ) | ≤ | S | . Pr o of. An y vertex of S that is no t in Γ( S ) is a connecting vertex of a 2-linked path in S . The endpoints o f that 2-linked path are in Γ( S ). Th us | Γ( S ) | ≥ 1. The r est follows from the fact that every vertex of Γ( S ) is a vertex o f S . Note that Γ( S ) contains hyper edges of size betw een 2 and k . F or each 2 ≤ i ≤ k , we deﬁne ℓ i to b e the nu mber of i -edges in Γ( S ). Lemma 50. If every vertex in H has de gr e e at le ast 2 then P k i =2 ( i − 1) ℓ i ≥ (1 + 1 2 k ) | V (Γ( S )) | . Pr o of. As we said a bove, every v ∈ V (Γ( S )) has the same degr ee in Γ( S ) a s it do es in H . Th us Γ ( S ) has minim um degre e at least 2. Consider any v of degree 2 in Γ( S ). Then v has degree 2 in H and hence canno t be the endp oint of a 2-linked path in S , unles s v lies in at lea st one hyp eredge of H containing more than 2 members of S . It follows that v lie s in at leas t o ne hyper edge o f Γ( S ) o f size greater than 2. Therefor e, at most P k i =3 iℓ i < k P k i =3 ℓ i vertices of Γ( S ) ha ve degree 2, a nd so letting Z denote the nu mber of vertices with deg ree a t leas t 3 in Γ( S ), w e have | V (Γ( S )) | ≤ Z + k k X i =3 ℓ i ≤ k Z + k X i =3 ℓ i ! . By the handshaking lemma, P k i =2 iℓ i = P v deg Γ( S ) ( v ). Therefore, 25 k X i =2 ( i − 1) ℓ i = 1 2 X v deg Γ( S ) ( v ) + k X i =2 ( i/ 2 − 1) ℓ i ≥ 1 2 X v deg Γ( S ) ( v ) + 1 2 k X i =3 ℓ i = X v 1 + X v 1 2 (deg Γ( S ) ( v ) − 2) + 1 2 k X i =3 ℓ i ≥ | V (Γ( S )) | + 1 2 Z + 1 2 k X i =3 ℓ i , since deg Γ( S )( v ) ≥ 2 for a ll v ≥  1 + 1 2 k  | V (Γ( S )) | . Let C b e the 2-core o f H = H k ( n, p ). W e will apply Lemma 50 w ith H = C to prov e: Lemma 51. Th er e exists α > 0 such that w.h.p. C has no non-empty linke d set of size less than αn . Lemma 45 follows immediately from Lemma 51 and Pro p osition 48 (since H k ( n, p ) contains only simple hyperedges). The pro of of Lemma 5 1 will b e r eminiscent of the pro of of Lemma 40, but s igniﬁcantly mo re complicated bec ause (i) we are working in the co nﬁguration mo del and (ii) where we had ℓ 2 2-edges in Lemma 51, we hav e ℓ 2 2-linked paths her e. Firs t, we provide a tec hnical lemma. Lemma 52. F or any inte gers a, t , given a set of a vertic es in H = H k ( n, p ) , with p = c/n k − 1 the pr ob ability that their t otal de gr e e exc e e ds tk ca is at most ( e/t ) act . Pr o of. Giv en a s et A of a v ertices , let E A denote the num b er of h yp eredges containing at least o ne member of A . The total degree in A is at most kE A . The n umber of potential edges in E A is at most a  n k − 1  < an k − 1 , and so E A is dominated from ab ov e by Bin( an k − 1 , c/n k − 1 ) a nd using  n z  ≤ ( ne/z ) z we get Pr  Bin( an k − 1 , c/n k − 1 ) > act  <  an k − 1 act   c n k − 1  act < ( e/ t ) act . Pr o of of L emma 51. By Corollar y 23, we can work in the conﬁgura tion mo del. Let D b e the degree sequence o f C . Recalling Deﬁnition 2 9, P rop osition 3 0 and our k ey Lemma 32, we ha ve w.h.p. (i) D has tota l degr ee γ n + o ( n ), where γ = µ Ψ r ( µ ), (ii) D has λ 2 n + o ( n ) vertices of degree 2, wher e λ 2 = e − µ µ 2 / 2, (iii) there exists ζ > 0 such that 2 ( k − 1) λ 2 < (1 − ζ ) γ . F or e ach a ≥ 1, let X a denote the n um b er of link ed sets S in C for which | Γ( S ) | = a and let X = P αn a =1 X a . Deﬁne L a = ( ( ℓ 2 , . . . , ℓ k ) :  1 + 1 2 k  a ≤ k X i =2 ( i − 1) ℓ i ≤  1 + 1 2 k  a + ( k − 1) ) . By Lemma 5 0, for any linked set S in C with | Γ( S ) | = a , ther e is some ( ℓ 2 , . . . , ℓ k ) ∈ L a so tha t Γ( S ) contains at le ast ℓ i i -edges for each i . 26 T o b ound E ( X a ), we b egin by cho osing a vertices, A ⊆ V ( C ) and sum over all t ≥ 0 o f the proba bilit y that t heir total deg ree in C lies in the r ange ( tk ca , ( t + 1) k ca ]. F or each t , w e upp er bound this las t proba bilit y by the probability that their total degree in H lies in ( tk ca, ∞ ]. Moreover, t o sum ov er all subsets A ⊆ V ( C ) we overcoun t by summing instead ov er all A ⊆ V ( H ), and us ing Lemma 52. Of course, if such a set is not a subset of C then the probability of it co n tributing to X a is zero, a nd so this provides a n upper bo und on E ( X a ). This yields:  n a  X t ≥ 0 ( e t ) tca . Given A , we sum ov er all poss ibilities for the v alues of ( ℓ 2 , . . . , ℓ k ) ∈ L a . F or e ach 2 ≤ i ≤ k , we choose ℓ i i -sets of vertex-copies belong ing to vertices of A . If the total degree of A is in ( tk ca, ( t + 1) k ca ] then the nu mber of c hoices for these ℓ i i -sets is at most  (( t + 1) k ca ) i i !  ℓ i /ℓ i ! < (( t + 1) k ca ) iℓ i ℓ i ! . Denote the ℓ 2 2-sets as { u 1 , w 1 } , . . . , { u ℓ 2 , w ℓ 2 } . F or each i = 1 , . . . , ℓ 2 , we select j i ≥ 0, the n umber of connecting v ar iables in the 2 -linked path from u i to w i in S , we choo se the j i degree tw o connecting v ar iables for that path, and we choo se one o f the tw o p ossible o rientations of the vertex-copies o f each of those connecting v ariables. Let J = j 1 + · · · + j ℓ 2 , be the num b er of connecting v ariables selected. Let L = λ 2 n + o ( n ) b e the num b er of degree 2 v ertices in C . Then the to tal num b er of choices for the connecting vertices and the orientations of their co pies is at most J Y i =1 2( L − i + 1) . Next, we apply Lemma 24 to bo und the pr obability that the ℓ 3 + · · · + ℓ k sets of size at le ast 3 all la nd in hyp eredges of the conﬁgur ation and that for each i = 1 , . . . , ℓ 2 , the ﬁrst pair in the 2- linked path, i.e., u i and the ﬁrst copy of the ﬁrst of the j i connecting v aria bles, lands in a hyperedg e of the conﬁguration. Note that ℓ 2 + · · · + ℓ k ≤ P k i =2 ( i − 1) ℓ i < 2 a + k − 1 < 2 a + o ( n ), b y the deﬁnition of L a . By assuming a < αn for some suﬃciently small α , we get γ n + o ( n ) − 2 a > 1 2 γ n . Therefor e, Lemma 24 yields that this probability is at most exp  k ( ℓ 2 + · · · ℓ k ) 2 1 2 γ n  k Y i =2  ( k − 1)( k − 2) · · · ( k − i + 1) ( γ n + o ( n )) i − 1  ℓ i < exp  8 k a 2 γ n  k Y i =2  k γ n  ( i − 1) ℓ i . F ollowing the ana lysis of Lemma 24, we have now exp osed ℓ 2 + · · · + ℓ k hyperedges o f the co nﬁguration. Let Λ b e the num ber o f unmatched vertex-co pies remaining . Since ℓ 2 + · · · + ℓ k < 2 a + k − 1, we hav e Λ ≥ γ n − 2 k a + o ( n ). If the o ther vertex-copies required for the 2-linked paths ar e still unmatched, then we contin ue; else w e halt obser ving that in this c ase, the set of choices made so far canno t lea d to a linked set on the chosen vertices. There are J pair s of vertex copies that ea ch need to b e in a hyperedg e of the conﬁgura tion in or der to complete the 2-link ed pa ths. F ollowing th e same argument as in Lemma 35, the probability of this happening is at most J Y i =1 k − 1 Λ − k ( i − 1) . Applying (iii) ab ove, and taking a < αn for α suﬃcien tly small in terms of γ , λ 2 , w e o btain: 2( k − 1) L Λ < 2( k − 1) λ 2 n + o ( n ) γ n − 2 k a + o ( n ) < 1 − ζ 2 . 27 Thu s, sinc e 2( k − 1) L ≤ Λ (b y the previous line) and k ≤ 2( k − 1), we hav e 2( k − 1)( L − ( i − 1)) Λ − k ( i − 1) < 1 − ζ 2 for each i , lea ding to E ( X a ) <  n a  X t ≥ 0 ( e t ) tca X ℓ 2 ,...,ℓ k ∈L a X j 1 ,...,j ℓ 2 ≥ 0 e 8 ka 2 / ( γ n ) k Y i =2 (( t + 1) k ca ) iℓ i ℓ i ! ! × k Y i =2  k γ n  ( i − 1) ℓ i ! J Y i =1 2( k − 1)( L − ( i − 1)) Λ − k ( i − 1) ! <  en a  a X t ≥ 0 ( e t ) tca X ℓ 2 ,...,ℓ k ∈L a e 8 ka 2 / ( γ n ) k Y i =2 ( k ca ) ℓ i ℓ i !  k 2 ca γ n  ( i − 1) ℓ i ( t + 1) iℓ i ! × X j 1 ,...,j ℓ 2 ≥ 0 (1 − ζ / 2) J . Since J = j 1 + · · · + j ℓ 2 , w e have P j 1 ,...,j ℓ 2 ≥ 0 (1 − ζ / 2) J =  P j ≥ 0 (1 − ζ / 2) j  ℓ 2 = (2 /ζ ) ℓ 2 , yielding: E ( X a ) <  en a  a e 8 ka 2 /γ n X ℓ 2 ,...,ℓ k ∈L a  k 2 ca γ n  P k i =2 ( i − 1) ℓ i k Y i =2 ( k ca ) ℓ i ℓ i ! ! ( 2 ζ ) ℓ 2 X t ≥ 0 ( e t ) tca ( t + 1) P k i =2 iℓ i . By our choice of L a ℓ 2 ≤ k X i =2 ( i − 1) ℓ i ≤ (1 + 1 2 k ) a + k − 1 , k X i =2 iℓ i ≤ 2 k X i =2 ( i − 1) ℓ i ≤ 3 a + 2 k . Thu s, w e obta in (2 /ζ ) ℓ 2 < Z a 1 for constant Z 1 = Z 1 ( c ) a nd, since ( e/ t ) tc ( t + 1) 3+2 k is decr easing for lar ge t , we ha ve X t ≥ 0 ( e/t ) tca ( t + 1) P k i =2 iℓ i < X t ≥ 0 ( e/t ) tca ( t + 1) 3 a +2 k < X t ≥ 0  ( e/t ) tc ( t + 1) 3+2 k  a 0. This yields E ( P √ n a =1 X a ) = o (1 ). Moreov er, for all α suﬃciently small, E ( X a ) < 2 − a . Therefore, E ( P a ≥ √ n X a ) = o (1) and, thus, E ( X ) = o (1). Therefore, w.h.p. there is no 2-linked set S with 1 ≤ | Γ( S ) | ≤ αn . The lemma follo ws fro m Pr op osition 49.  28 References [1] D. Ac hlioptas, P . Beame, and M. Molloy . A sharp thr eshold in pr o of c omplexity yields lower b ounds for satisﬁability se ar ch. J. Comput. Sys t. Sci., 68 (2), 2 38–26 8 (20 04). [2] D. Ac hlioptas and A. Co ja-Oghla n. Algorithmic ba rriers from phase transitions . In 49th Annual IEEE Symp osium on F oundations of Computer Scienc e, FOCS 2008, Octob er 25-28, 2008, Philadelphi a, P A, USA , page s 79 3–802 . IEE E Computer So ciety , 200 8. [3] D. Achlioptas, A. Co ja-O ghlan and F. Ricci-T ersenghi On the solution-sp ac e ge ometry of ra ndom c on- str aint satisfaction pr oblems. Random Str uct. Algo rithms, 38 (3), 2 51-26 8 (2011). [4] D. Ac hlioptas and F. Ricc i-T ers enghi R andom F ormulas Have F r ozen V ariables. SIAM J. Comput., 39 (1), 26 0-280 (200 9). [5] E.R. Berlek amp, R.J. McEliec e, and H.C.A. v an Tilb or g. On the inher ent intr actability of c ertain c o ding pr oblems. IEEE T rans. Inform. Theor y 24 , 384 386 (1978 ). [6] B. Bo llob´ as. A pr ob abilistic pr o of of an asymptotic formula for the n umb er of lab el le d r e gular gr aphs. Europ. J. Combinatorics 1 311-31 6 (19 80). [7] H.S. Connamacher and M. Molloy . The satisﬁa bilit y thres hold for a s eemingly intractable rando m constraint satisfaction pr oblem. SIAM J. Discr ete Math. , 26(2):768 –800, 2012. [8] C. Co op er. The c or es of r andom hyp er gr aphs with a given de gr e e se quenc e. Random Structures Algo- rithms 25 (2004) 353 - 375. [9] H. Daud´ e, M. M ´ ezard, T. Mora, a nd R. Zecchina. Pai rs of SA T Assignments and Clustering in R andom Bo ole an F ormulae. Theoretical Computer Science, 393 (1-3), 2 60-27 9 (20 08). [10] M. Dietzfelbinger , A. Go erdt, M. Mitzenmacher, A. Mo n tanar i, R. Pagh and M. Rink. Tight thresh- olds for cuck o o hashing via XORSA T. In Automata, L anguages and Pr o gr amming, 37th International Col lo qu ium, ICALP 2010, Bor de aux , F r anc e, July 6-10, 2010, Part I , pa ges 213-225 . Springe r, 20 10. [11] O . Dubois and J . Mandler. The 3-X ORSA T threshold. In 43r d Symp osium on F oundations of Computer Scienc e (F OCS 2002 ), 16-19 Novemb er 2002, V anc ouver, BC, Canada , pages 769-77 8. IEEE Computer So ciety , 2002 . [12] M. Guidetti and A.P . Y o ung. Complexity of sever al c onstr aint-satisfaction pr oblems u sing the heuristic classic al algorithm WalkSA T. Phys. Rev. E , 84 (1), 0 11102 , July 201 1. [13] M. Ibrahimi, Y. Kanor ia, M. K raning, and A. Mon tanari. The set of solutio ns of ra ndom xorsat formulae. In Pr o c e e dings of the Twenty-Thir d Annual AC M-SIAM Symp osium on Discr ete Algorithms, SODA 2012, Kyoto, Jap an, Janu ary 17-19, 2012 , pages 7 60–77 9. SIAM, 20 12. [14] S. Jans on, T. Luczak and A. Ruci ´ nski. Random Gra phs. Wiley , New Y or k (200 0). [15] J .H.Kim. Poisson cloning mo del for r andom gr aphs. arXiv:0805 .4133 v1 [16] T. L uczak. Comp onent b ehavio ur ne ar the critic al p oint of the r andom gr aph pr o c ess . Rand. Struc. & Alg. 1 (1990 ), 287 - 31 0. [17] M. M´ ezar d and A. Montanari. Information, Physics, and Computation . O xford Universit y Press, Inc., New Y o rk, NY, USA, 2009. [18] M. M´ ezard, T. Mora , a nd R. Zecchina Clustering of S olutions in t he R andom S atisﬁability Pr oblem. Phys. Rev . Lett., 94 (19), 19 7205 (200 5). 29 [19] M. M´ ezar d, G. Parisi, and R. Z ecchina Analytic and A lgorithmic Solution of R andom Satisﬁability Pr oblems. Science, 297 , 81 2 - 815 (20 02). [20] M. M´ e zard, F. Ricci-T ersenghi, and R. Zecchina. Alternative solutions to dilute d p -spin mo dels and XORSA T pr oblems. J . Stat. Phys. 111 , 505 - 533, (20 03). [21] M. Mo lloy Cor es in r andom hyp er gr aphs and b o ole an formulas. Random Structures and Algorithms 27 , 124 - 135 (20 05). [22] M. Molloy and B. Reed. A critic al p oint for r andom gr aphs with a given de gr e e se quenc e. Random Structures and Algorithms 6 161 - 180 (199 5). [23] M. Molloy and M. Salav a tipo ur. The r esolution c omplexity of r andom c onstr aint satisfaction pr oblems. SIAM J . Comp. 37 , 8 95 - 922 (2007). [24] B . Pittel. On t r e es c ensu s and t he giant c omp onent in sp arse r andom gr aphs. Random Structures and Algorithms 1 (1990 ), 311 - 342 . [25] B . Pittel, J. Sp encer a nd N. W or mald. Sudden emer genc e of a giant k -c or e in a r andom gr aph. J. Com b. Th. B 67 , 1 11 - 151 (1996). 30

The solution space geometry of random linear equations

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment