A Fast Approximation Algorithm for the Minimum Balanced Vertex Separator in a Graph

A F ast Appro ximation Algorithm for the Minim um Balanced V ertex Separator in a Graph Vladimir Kolmogoro v Institute of Science and T ec hnology Austria (IST A) vnk@ist.ac.at Jac k Spalding-Jamieson Indep enden t jacksj@uwaterloo.ca Abstract W e presen t a family of fast pseudo-appro ximation algorithms for the minim um balanced v er- tex separator problem in a graph. Giv en a graph G = ( V , E ) with n v ertices and m edges, and a (constan t) balance parameter c ∈ (0 , 1 / 2), where G has some (unkno wn) c -balanced vertex sep- arator of size OPT c , w e giv e a (Monte-Carlo randomized) algorithm running in O ( n O ( ε ) m 1+ o (1) ) time that produces a Θ(1)-balanced v ertex separator of size O (OPT c · p (log n ) /ε ) for an y v alue ε ∈ [Θ(1 / log( n )) , Θ(1)]. In particular, for an y function f ( n ) = ω (1) (including f ( n ) = log log n , for instance), w e can pro duce a vertex separator of size O (OPT c · √ log n · f ( n )) in time O ( m 1+ o (1) ). Moreo ver, for an arbitrarily small constan t ε = Θ(1), our algorithm also ac hieves the b est-kno wn approximation ratio for this problem in O ( m 1+Θ( ε ) ) time. The algorithms are based on a semidefinite programming (SDP) relaxation of the problem, whic h we solv e using the Matrix Multiplicative W eigh t Up date (MMWU) framework of Arora and Kale. Our oracle for MMWU uses O ( n O ( ε ) p olylog( n )) almost-linear time maxim um-flow computations, and would b e sp ed up if the time complexity of maximum-flo w improv es. 1 In tro duction P artitioning a graph in to smaller pieces is a fundamen tal problem that arises in many fields. One kind of graph partition with widespread applications in algorithms is the vertex separator : A small set of v ertices whose remov al disconnects the graph into small comp onents. More formally , let G = ( V , E ) b e a graph with n = | V | vertices. Let w : V → N b e a set of in teger weigh ts for the v ertices. Let c ∈ (0 , 1 2 ) b e a constan t. F or an y subset S ⊂ V , w e let w ( S ) denote P s ∈ S w ( s ). Define a c -balanced v ertex separator to b e a subset of vertices C ⊂ V suc h that V \ C can b e partitioned into tw o sets A and B with no edges b etw een them and max {| A | , | B |} ≤ (1 − c ) n . The size of the vertex separator is w ( C ). In many applications, w ( · ) = 1, but we present our results in this more general form where the cost of the separator may v ary . Small vertex separators are known to exist for man y classes of graphs with uniform weigh ts w ( · ) = 1. Planar graphs are known to admit 1 3 -balanced vertex separators of size O ( √ n ) [25, 16], gen us- g graphs are kno wn to admit 1 3 -balanced v ertex separators of size O ( √ g n ) [8], and similar results are kno wn for minor-free graphs [11], geometric graphs [20, 24], induced-minor-free graphs [13], and other classes [19, 6]. These small separators also hav e numerous applications. T o name a few, they naturally allow devising divide and conquer algorithms, dynamic programming algorithms, sub exp onential algorithms for NP-hard problems on special graphs, and maximum matc hing algorithms [21, 17, 13]. Although many classes of graphs are kno wn to admit small separators, algorithms for con- structing them are often quite slow. Th us, we naturally turn to the question of appro ximation algorithms: Given a graph G , can we appro ximate the minim um c -balanced v ertex separator? It turns out that this is essentially imp ossible as-stated: It is NP-hard to appro ximate the mini- m um c -balanced vertex separator up to an additive appro ximation of O ( n 1 2 − ϵ ) for an y ϵ > 0 [4]. Instead, we turn to pseudo -approximations (also called bi-criteria appro ximations): F or a graph 1 G whose minim um c -balanced vertex separator has size OPT c , we can pro duce a Θ( c )-balanced v ertex separator with size close to OPT c . Our main result is as follo ws: Theorem 1. F or any ε ∈ [Θ(1 / log n ) , Θ(1)] and any c onstant c ∈ (0 , 1 / 2) , ther e is a Monte- Carlo r andomize d algorithm (suc c e e ding w.h.p.) that, given a gr aph G = ( V , E ) and inte ger vertex weights w : V → N , finds a Θ(1) -b alanc e d vertex sep ar ator of size at most O (OPT c · p (log n ) /ε ) , wher e OPT c is the minimum size of a c -b alanc e d vertex sep ar ator. The algorithm runs in time O  n O ( ε ) · T maxflow ( n, m ) · polylog ( n max i w ( i ))  , wher e T maxflow ( n, m ) is the time to solve a max- flow pr oblem on a gr aph with n vertic es and m e dges with c ap acities al l p oly ( n ) . By applying the almost-linear time maxim um flo w algorithm of Chen, Kyng, Liu, P eng, Guten- b erg, and Sac hdev a [5], w e know that T maxflow = O ( m 1+ o (1) ). Separator Size Running Time Ref Notes O (OPT · log n ) p oly( n ) [15] O (OPT · √ log n ) Time to solv e O ( n ) SDPs [7, 1] Best known general SDP solv- ing time is e O ( √ n ( mn 2 + m ω + n ω )) [9]. e O (OPT 2 ) e O (OPT 3 m ) [3] Both runtime and appro xima- tion ratio dep end on OPT. O  OPT · q log n ε  O ( n 1+ O ( ε ) m 1+ o (1) ) [14] Uses MMWU to solv e an SDP relaxation in almost-linear time. Requires O ( n ) runs for balance, resulting in the extra factor of n . W orks for any ε ∈ [1 / Θ(log n ) , Θ(1)]. e O (OPT) e O ( m 1+ o (1) ) [18] Unpublished w ork. Details for applying to balanced v ertex- separator problem are not pro- vided. Requires almost-linear time max-flo w [5]. O  OPT · q log n ε  O ( n O ( ε ) m 1+ o (1) ) Ours W orks for an y ε ∈ [1 / Θ(log n ) , Θ(1)]. T able 1: Previous pseudo-approximation algorithms for the balanced v ertex separator problem. OPT denotes the size of the smallest 2 3 -balanced v ertex separator. The notation e O hides p olylog- arithmic factors. Although a long series of p olynomial-time pseudo-appro ximation algorithms exist for the bal- anced vertex separator problem (see T able 1), the b est approximation ratio attainable in almost- linear time is a (large) p olylogarithmic ratio, informally stated without pro of in the conclusion of an unpublished result b y Louis [18]. Consequently , the following immediate corollary to Theorem 1 (using ε = Θ(1 /f ( n )) for a slo w-growing function f ( n )) is a significant step forward: Corollary 2. L et f ( n ) = ω (1) b e a function (e.g., f ( n ) = log log n ). Ther e is a Monte-Carlo r andomize d algorithm (suc c e e ding w.h.p.) that, given a gr aph G = ( V , E ) and p olynomial ly-b ounde d inte ger vertex weights w : V → [ O ( n O (1) )] , finds a Θ(1) -b alanc e d vertex sep ar ator of size at most O (OPT c · √ log n · f ( n )) . The algorithm runs in O ( m 1+ o (1) ) time, plus the time to evaluate f ( n ) . 2 On the other hand, the b est-known appro ximation factor for the balanced v ertex separator problem is O ( √ log n ) [7], and w e can match that result with an algorithm that gets arbitrarily close to almost-linear time: Corollary 3. F or any c onstant δ > 0 , ther e is a Monte-Carlo r andomize d algorithm (suc c e e ding w.h.p.) that, given a gr aph G = ( V , E ) and p olynomial ly-b ounde d inte ger vertex weights w : V → [ O ( n O (1) )] , finds a Θ(1) -b alanc e d vertex sep ar ator of size at most O δ ( √ log n · OPT c ) . The algorithm runs in O ( m 1+ δ ) time. Finally , b y c ho osing ε = Θ(1 / log n ), we can attain a time complexity that is entirely dependent on T maxflow ( n, m ): Corollary 4. Ther e is a Monte-Carlo r andomize d algorithm (suc c e e ding w.h.p.) that, given a gr aph G = ( V , E ) and p olynomial ly-b ounde d inte ger vertex weights w : V → [ O ( n O (1) )] , finds a Θ(1) - b alanc e d vertex sep ar ator of size at most O (OPT c · log n ) . The algorithm runs in O ( T maxflow ( n, m ) · p olylog( n )) time. This last corollary will b e useful if T maxflow ( n, m ) is ever impro ved to b e near-linear. 1.1 T echnical Ov erview Our tec hniques are comprised of three elements: A con v ex relaxation, a regret-minimization frame- w ork for certain con vex relaxations, and an efficient implementation of an imp ortan t piece for the framew ork. First, w e give a semidefinite programming relaxation of the minimum c -balanced v ertex sepa- rator problem in section 2. This relaxation is similar to the relaxation of the minim um c -balanced e dge separator problem used by Arora and Kale [2]. Directly solving a semidefinite program would be to o slow, so we instead deplo y the Matrix Multiplicativ e W eight Up date framew ork of Arora and Kale for approximately solving com binato- rial optimization problems [2], whic h w e review in section 2.1. Deplo ying this framework efficien tly also requires random pro jections (which is wh y our algorithm is randomized), whic h w e discuss in section 3. In order to implement the Matrix Multiplicative W eight Update framework, w e must pro vide an “oracle” that either pro vides a sp ecial “feedbac k matrix”, or terminates early and provides an appro ximate com binatorial solution. Implemen ting this oracle with our desired approximation factor requires building on the metho ds of Sherman, Lau, T ung, and W ang, and Kolmogoro v [22, 14, 12], while also calling a subroutine for single-commo dit y maximum flow. The implementation of the oracle is detailed in section 4. 2 Preliminaries W e will no w define the main con vex relaxation w e will work with. Let ξ := 9 4 c 2 , let S := { S ⊆ V : | S | ≥  1 − c 4  n } , and let P be the set of sequences p = ( p 0 , . . . , p ℓ ( p ) ) of distinct no des in G . W e use the notation p = ( p 0 , . . . , p ℓ ( p ) ) for the sequence of v ertices along a path p ∈ P . W e consider the follo wing relaxation of the c -balanced vertex separator problem: 3 min X i ∈ V w ( i ) · x i (1a) x i + x j ≥ || v i − v j || 2 x i + x j − L ij • X ≥ 0 ∀ ij ∈ E (1b) || v i || 2 = 1 X ii = 1 ∀ i ∈ V (1c) ℓ ( p ) X j =1 || v p j − v p j − 1 || 2 ≥ || v p ℓ ( p ) − v p 0 || 2 T p • X ≥ 0 ∀ p ∈ P (1d) X i,j ∈ S : i n/ 2) or ( ˆ A, ˆ B ) = ( V \ B , B ) (if | B | > n/ 2). In all three cases w e hav e | ˆ A | ≥ cn and | ˆ B | ≥ cn , since c < 1 / 2. Set x i = ( 0 if i ∈ A ∪ B 4 if i ∈ C and v i = ( − 1 if i ∈ ˆ A +1 if i ∈ ˆ B . Checking conditions (1b), (1c), (1d), (1f) is straightforw ard. (Note that for eac h edge ij ∈ E with v i  = v j w e must ha ve either i ∈ C or j ∈ C since E has no edges b et ween A and B , and hence x i + x j ≥ 4 ≥ || v i − v j || 2 .) Let us show (1e). Consider a set S ⊆ V with | S | ≥ (1 − c 4 ) n . W e hav e | ˆ A ∩ S | ≥ | ˆ A | − c 4 n ≥ 3 c 4 n , and similarly for ˆ B ∩ S . Therefore, X i,j ∈ S : i 0 ther e exists an algorithm that do es the fol lowing: given values γ ∈ (0 , 1 2 ) , λ max > 0 , τ = O ( n 3 / 2 ) and (implicit) matrix A ∈ R n × n of sp e ctr al norm || A || ≤ λ max , it c omputes a matrix ˜ V ∈ R d × n with c olumn ve ctors ˜ v 1 , . . . , ˜ v n of dimension d = O ( log n γ 2 ) such that matrix ˜ X = ˜ V T ˜ V has tr ac e n , and with pr ob ability at le ast 1 − n − c , one has | || ˜ v i || 2 − || v i || 2 | ≤ γ ( || ˜ v i || 2 + τ ) ∀ i (3a) | || ˜ v i − ˜ v j || 2 − || v i − v j || 2 | ≤ γ ( || ˜ v i − ˜ v j || 2 + τ ) ∀ i, j (3b) wher e v 1 , . . . , v n ar e the c olumns of a Gr am de c omp osition of X = n · exp( A ) Tr (exp( A )) . The c omplexity of this algorithm e quals the c omplexity of c omputing k d matrix-ve ctor pr o ducts of the form A · u , u ∈ R n , wher e k = O (max { λ 2 max , log n 5 / 2 τ } ) . If we use this theorem inside Algorithm 1, then the matrices A will ha ve the form A = η P t − 1 r =1 N ( r ) ; their sp ectral norm will b e b ounded b y η ρT . Therefore, we can set λ max = η ρT = Θ( ρn log n α ) in Theorem 7. P arameters γ and τ will b e sp ecified later. 6 W e will write ( v 1 , . . . , v n ) ≈ γ ,τ ( ˜ v 1 , . . . , ˜ v n ) if conditions (3) hold. W e now need to sho w ho w to solv e the following problem. Input : (unobserved) matrix V ∈ R n × n with columns v 1 , . . . , v n ∈ R n and (observ ed) matrix ˜ V ∈ R d × n with columns ˜ v 1 , . . . , ˜ v n ∈ R d suc h that ( v 1 , . . . , v n ) ≈ γ ,τ ( ˜ v 1 , . . . , ˜ v n ) and Tr ( X ) = Tr ( ˜ X ) = n where X = V T V and ˜ X = ˜ V T ˜ V . Output : either (i) matrix N of the form N = diag ( y ) + P p f p T p + P S z S K S − P ij λ ij L ij where f p ≥ 0, z S ≥ 0, P i y i + ξ n 2 P S z S ≥ α , λ ij are non-negativ e v ariables with deg ( G λ ) ≤ w , and N • X ≤ 0; or (ii) a Θ(1)-balanced v ertex cut of v alue at most κα . A procedure that solves the problem ab ov e will be called an “ Oracle ”, and the sp ectral norm || N || of matrix N will be called the width of the oracle. 4 Oracle implemen tation In this section, w e will show how to implement the oracle used for MMWU. That is, w e will solve the problem sp ecified at the end of the previous section. First, w e will handle an easy case that returns early if triggered. Let us denote S = { i ∈ V : || ˜ v i || 2 ≤ 4 /c } . W e ha ve P i ∈ V || ˜ v i || 2 = Tr ( ˜ V T ˜ V ) = n and th us | S | ≥ (1 − c/ 4) n . Prop osition 8. Supp ose that K S • ˜ X < ξ n 2 4 . Then setting y i = − α n for al l i ∈ V , z S = 2 α ξ n 2 , z S ′ = 0 for al l S ′  = S , and λ ij = 0 gives a valid output of the or acle with width ρ = O ( α n ) assuming that p ar ameters τ , γ in The or em 7 satisfy γ ≤ 1 2 and τ ≤ ξ 2 . Pr o of. Denote z ij = || v i − v j || 2 and ˜ z ij = || ˜ v i − ˜ v j || 2 . W e know that ˜ Z := P ij ˜ z ij = K S • ˜ X < ξ n 2 4 where the sum is ov er i, j ∈ S with i < j . Also, | z ij − ˜ z ij | ≤ γ ( ˜ z ij + τ ) for all i, j . This implies that K S • X = X ij z ij < ˜ Z + γ ˜ Z + n 2 2 γ τ ≤ (1 + γ ) ξ n 2 4 + n 2 2 γ τ ≤ ξ n 2 2 Note that N = − α n I + 2 α ξ n 2 K S . W e ha ve P i y i + ξ n 2 P S ′ z S ′ = n · ( − α n ) + ξ n 2 · 2 α ξ n 2 = α and N • X = − α n I • X + 2 α ξ n 2 ( K S • X ) ≤ − α n · n + 2 α ξ n 2 · ξ n 2 2 = 0, as desired. Also, || N || ≤ ||− α n I || + || 2 α ξ n 2 K S || = O ( α n ). F rom no w on we make the following assumption. Assumption 1. || ˜ v i || 2 ≤ 4 /c for al l i ∈ S , | S | ≥ (1 − c/ 4) n and P i,j ∈ S || ˜ v i − ˜ v j || 2 = K S • ˜ X ≥ ξ n 2 4 . Let us set y i = α n for all i ∈ V and z S ′ = 0 for all S ′ , then P i y i + ξ n 2 P S ′ z S ′ = α . Note that N = α n I + P p f p T p − P ij λ ij L ij and α n I • X = α n Tr ( X ) = α , so condition N • X ≤ 0 is equiv alent to ( P p f p T p − P ij λ ij L ij ) • X ≤ − α . With a v ariable substitution, our goal thus b ecomes as follo ws. Find either (i) a matrix N of the form N = P p f p T p − P ij λ ij L ij wher e f p ≥ 0 , λ ij ≥ 0 , deg ( G λ ) ≤ w , and N • X ≤ − α ; or (ii) a Θ(1) -b alanc e d vertex cut of value at most κα . In the remainder of this section we describ e ho w to solve this problem. T o simplify notation, w e will assume that vectors ˜ v i for i ∈ V are unique, and rename the no des in V so that ˜ v x = x for eac h x ∈ V . Th us, we no w hav e V ⊆ R d . The “true” vector in R n corresp onding to x ∈ V is still denoted as v x . 7 4.1 Pro cedure Matching ( u ) The main building blo ck of the oracle is a pro cedure that takes vector u ∈ R d and either outputs a directed matching M on no des V or terminates the oracle. It works as follo ws ( c ′ , ∆ , σ, β are p ositiv e constants that will b e sp ecified later): Algorithm 2: Matching ( u ). 1 compute w x = ⟨ x, u ⟩ for each x ∈ S 2 sort { w x } x ∈ V , let A, B b e subsets of S with | A | = | B | = 2 c ′ n con taining no des with the least and the greatest v alues of w x , resp ectiv ely 3 construct directed graph G ′ with no des V ′ = { x, ¯ x : x ∈ V } ∪ { s, t } as follo ws: (i) for eac h x ∈ V add edge ( x, ¯ x ) of capacit y w ( x ) 2 ; (ii) for eac h { x, y } ∈ E add edges ( ¯ x, y ), ( ¯ y , x ) of infinite capacit y; (iii) for each x ∈ A add edge ( s, x ) of capacit y β ; (iv) for each y ∈ B add edge ( ¯ y , t ) of capacit y β 4 compute maximum s - t flow f ′ and the corresp onding minim um s - t cut ( S cut , T cut ) in G ′ with minimal | T cut | 5 if capacity of the cut is less than c ′ nβ then return set U = { x ∈ V : x ∈ S cut , ¯ x ∈ T cut } and terminate the oracle 6 compute flo w decomp osition of f ′ . F or each path p ′ = ( s, x 1 , ¯ x 1 , . . . , x k , ¯ x k , t ) carrying flo w f ′ p ′ define path p = ( x 1 , . . . , x k ) (with x 1 ∈ A, x k ∈ B ) and set f p = f ′ p ′ . F or each pair x, y ∈ V set d xy = P p =( x,...,y ) f p . 7 if P x ∈ A,y ∈ B d xy || x − y || 2 ≥ 2 α , then set λ xy = f ′ ¯ xy + f ′ ¯ yx for eac h { x, y } ∈ E , return v ariables λ xy and matrix N = −L ( D ) where D is the symmetric matrix with D xy = d xy if x ∈ A, y ∈ B (and other en tries zeros). T erminate the oracle. 8 let M all = { ( x, y ) ∈ A × B : d xy > 0 , w y − w x ≥ σ } and M short = { ( x, y ) ∈ M all : || x − y || 2 ≤ ∆ } 9 pick maximal matching M ⊆ M short and return M Note that in line 6 there are at most O ( m ) paths p ′ = ( s, x 1 , . . . , ¯ x k , t ), and for eac h suc h path it suffices to compute only the endp oints x 1 , ¯ x k . These computations can b e done in O ( m log n ) time using dynamic trees [23]. Lemma 9. If the algorithm terminates at line 5 then the r eturne d set U is a c ′ -b alanc e d vertex sep ar ator with w ( U ) ≤ 2 c ′ nβ . Pr o of. W e ha ve c ′ nβ ≥ cost ( S cut , T cut ) ≥ P x ∈ U w ( x ) 2 and hence w ( U ) ≤ 2 c ′ nβ . Let us sho w that U is a c ′ -balanced vertex separator. There are no no des x ∈ V with x ∈ T cut , ¯ x ∈ S cut (otherwise w e could reassign x to S cut without increasing the cost of the cut, which would con tradict the minimalit y of T cut ). Thus, X ⊔ Y ⊔ U is a partitioning of V , where X = { x ∈ V : x, ¯ x ∈ S cut } and Y = { x ∈ V : x, ¯ x ∈ T cut } . There are no edges in G b etw een X and Y , since otherwise there w ould b e an edge of infinite capacity from S cut to T cut in G ′ . W e ha ve c ′ nβ ≥ cost ( S cut , T cut ) ≥ P x ∈ Y ∩ A β , and hence | Y ∩ A | ≤ c ′ n . Condition | A | = 2 c ′ n then implies that | Y | ≤ (1 − c ′ ) n . By a symmetric argumen t, | X | ≤ (1 − c ′ ) n . Lemma 10. Supp ose that p ar ameters τ , γ in e q. (3) satisfy τ ≤ 2 , γ ≤ α (32 /c +2 τ ) c ′ nβ . If the algorithm terminates at line 7, then the r eturne d variables λ ij and matrix N ar e valid output of the or acle. Matrix N has at most O ( m ) non-zer o entries, and its sp e ctr al norm is at most β . Pr o of. Consider x ∈ V . There are at most w ( x ) 2 units of flow en tering ¯ x (through edge ( x, ¯ x )), thus at most w ( x ) 2 units of flow are leaving ¯ x through edges ( ¯ x, y ), y ∈ V . Similarly , at most w ( x ) 2 units of flo w are lea ving x (through edge ( x, ¯ x )), thus at most w ( x ) 2 units of flo w are en tering x through edges ( ¯ y , x ), y ∈ V . These t wo facts imply that P y : { x,y }∈ E λ xy = P y : { x,y }∈ E f ′ ¯ xy + P y : { x,y }∈ E f ′ x ¯ y ≤ w ( x ) 2 + w ( x ) 2 = w ( x ). 8 Consider path p ′ = ( s, x 1 , ¯ x 1 , . . . , x k , ¯ x k , t ) carrying flow f ′ p ′ = f p , where p = ( x 1 , . . . , x k ). The con tribution of this flo w to matrix P p f p T p −L ( G λ ) is f p · T p − f p L ( G p ) = − f p L ( x 1 ,x k ) . Summing this expression o ver all paths p ′ in the flow decomp osition gives the matrix N defined at line 7. Th us, w e indeed hav e N = P p f p T p − L ( G λ ). It remains to s ho w that N • X ≤ − α . By Assumption 1, || x || 2 ≤ 4 /c for each x ∈ S , and hence || x − y || 2 ≤ 16 /c for each x, y ∈ S . Since ( v 1 , . . . , v n ) ≈ γ ,τ ( ˜ v 1 , . . . , ˜ v n ), we get || x − y || 2 − || v x − v y || 2 ≤ γ ( || x − y || 2 + τ ) ≤ α (32 /c +2 τ ) c ′ nβ (16 /c + τ ) = α 2 c ′ nβ . W e can th us write − N • X = X x ∈ A,y ∈ B d xy || v x − v y || 2 ≥ X x ∈ A,y ∈ B d xy ( || x − y || 2 − α 2 c ′ nβ ) ≥ X x ∈ A,y ∈ B d xy || x − y || 2 − α 2 c ′ nβ X x ∈ A X y ∈ B d xy ≥ 2 α − α 2 c ′ nβ · | A | · β = α The maxim um degree of D is at most β , and hence || N || = ||L ( D ) || ≤ β . W e will write u ∼ N to indicate that u is a random vector in R d with Gaussian indep enden t comp onen ts u i ∼ N (0 , 1). Notation Pr u [ · ] will mean the probabilit y under distribution u ∼ N . W e will assume that if Matching ( u ) terminates at line 5 or 7 then it returns an em pt y matching. Th us, we alwa ys ha ve Matching ( u ) ⊆ V × V and | Matching ( u ) | ≤ | V | . Lemma 11. Supp ose that β ≥ 6 α c ′ n ∆ . Ther e exist p ositive c onstants c ′ , σ, δ for which either (i) E u | Matching ( u ) | ≥ δ n , or (ii) Algorithm 2 for u ∼ N terminates at line 5 or 7 with pr ob ability at le ast Θ(1) . Pr o of. The pro of is v ery similar to that of [12, Lemma 3.4(b)]. Assume that condition (i) is false. By a standard argument, Assumption 1 implies the follo wing: there exist constants c ′ ∈ (0 , c ) and σ > 0 suc h that with probability at least Θ(1) Algorithm 2 reac hes line 9 and we ha ve w y − w x ≥ σ for all x ∈ A , y ∈ B (see [10, Lemma 14]). Supp ose that this ev ent happ ens. W e claim that in this case | M | ≥ 1 3 c ′ n . Indeed, supp ose this is false. Let A ′ ⊆ A and ¯ B ′ ⊆ ¯ B def = { ¯ y : y ∈ B } b e the sets of no des inv olv ed in M (with | A ′ | = | ¯ B ′ | = | M | = k ). The total v alue of flow from A to ¯ B is at least c ′ β n (otherwise we would hav e terminated at line 5). The v alue of flow leaving A ′ is at most | A ′ | · β ≤ 1 3 c ′ β n . Similarly , the v alue of flow en tering ¯ B ′ is at most | ¯ B ′ | · β ≤ 1 3 c ′ β n . Therefore, the v alue of flow from A − A ′ to ¯ B − ¯ B ′ is at least c ′ β n − 2 · 1 3 c ′ β n = 1 3 c ′ β n . F or each edge ( x, y ) ∈ M all with x ∈ A − A ′ , ¯ y ∈ ¯ B − ¯ B ′ w e hav e || x − y || 2 > ∆ (otherwise M would not b e a maximal matc hing in M short ). Therefore, X p : p =( x,...,y ) f p || x − y || 2 ≥ X p : p =( x,...,y ) x ∈ A − A ′ , ¯ y ∈ ¯ B − ¯ B ′ f p || x − y || 2 ≥ 1 3 c ′ β n · ∆ ≥ 2 α But then the algorithm should ha ve terminated at line 7 - a contradiction. Let us define c ′ , σ, δ as in Theorem 11, and set ∆ = r ε log n (4) β ∈ [ β 0 , 2 β 0 ] , β 0 = 6 α c ′ n ∆ (5) The size of set U in Lemma 9 is then w ( U ) ≤ 2 c ′ nβ = Θ  α p (log n ) /ε  , which corresp onds to an O  p (log n ) /ε  pseudo-appro ximation algorithm. 9 4.2 Chaining algorithm The remaining part is iden tical to the algorithm in [12]. Assume that case (i) in Lemma 11 holds (otherwise calling Matching ( u ) for u ∼ N will terminate the oracle after O (1) exp ected calls). Assume also that Matching ( · ) is skew-symmetric , i.e. Matching ( − u ) is obtained from Matching ( u ) b y rev ersing edge orientations. (This can b e easily enforced algorithmically). The idea is to use pro cedure Matching ( · ) to find man y “violating paths”, i.e. paths p = ( p 0 , . . . , p ℓ ( p ) ) that satisfy ℓ ( p ) X j =1 || p j − p j − 1 || 2 ≤ || p ℓ ( p ) − p 0 || 2 − ∆ (6) F or a set of directed paths M let M violating b e the set of paths p ∈ M that are either violating or con tain a violating path q as a subpath ( p = ( . . . , q , . . . )). F or t wo sets of paths M , M ′ define M ◦ M ′ = { ( p, q ) : p ∈ M , q ∈ M ′ , endpoint ( p ) = startpoint ( q ) } Theorem 12 ([22, 12]) . L et Matching ( u ) b e a pr o c e dur e that for given u ∈ R d r eturns a dir e cte d matching on V in which every e dge ( x, y ) satisfies ⟨ y − x, u ⟩ ≥ σ and || x − y || 2 ≤ ∆ . Assume that it is skew-symmetric and E u [ | Matching ( u ) | ] = Ω( n ) . Ther e exists an efficiently samplable distribution D of ve ctors u = ( u 1 , . . . , u K ) with K = O (1 / ∆) such that set M ( u ) = Matching ( u 1 ) ◦ . . . ◦ Matching ( u K ) satisfies E u ∼D | M ( u ) violating | ≥ e − Θ( K 2 ) n . Set K = Θ(∆ log n ) = Θ  √ ε log n  , then indeed K = O (1 / ∆) = O  p (log n ) /ε  since ε = O (1). Let us run the algorithm in Theorem 12 un til w e get | M ( u ) | ≥ e − Θ( K 2 ) n = n 1 − Θ( ε ) . Since w e alw ays hav e | M ( u ) | ≤ n , this will happ en after n/n 1 − Θ( ε ) = n Θ( ε ) runs in exp ectation (or K n Θ( ε ) = n Θ( ε ) calls to Matching ( · )). Once w e ha ve a set P of violating paths with | P | ≥ n 1 − Θ( ε ) , the output of the oracle is constructed as follows. Set λ ij = 0 for all { i, j } ∈ E . Set f p = 2 α | P | ∆ for all p ∈ P , and f p = 0 for all other paths. Then N = P p f p T p − L ( G λ ) = 2 α | P | ∆ ( L ( F ) − L ( D )) where multigraph F is the union of paths in P and m ultigraph D is the union of edges { startpoint ( p ) , endpoint ( p ) } o v er p ∈ P . Note that the complexit y of the oracle is dominated b y n Θ( ε ) maxim um flo w computations. Lemma 13. Supp ose that p ar ameters τ , γ in e q. (3) satisfy τ < 2 and γ ≤ ∆ 20( K +1) . Then variables λ ij and matrix N define d ab ove ar e valid output of the or acle. Matrix N has at most O ( n √ log n ) non-zer o entries, and its sp e ctr al norm is at most α n 1 − Θ( ε ) . Pr o of. W e need to show that N • X ≤ − α . Using eq. (6) and the fact that ( v 1 , . . . , v n ) ≈ γ ,τ ( ˜ v 1 , . . . , ˜ v n ), w e conclude that for any p ∈ P we ha v e ℓ ( p ) X j =1 || v p j − v p j − 1 || 2 ≤ || v p ℓ ( p ) − v p 0 || 2 − 1 2 ∆ T p • X ≤ − 1 2 ∆ (using the same argument as in the pro of of Theorem 10). Therefore, N • X ≤ | P | · 2 α | P | ∆ · ( − 1 2 ∆) = − α , as desired. By construction, matrix N has at most O ( | P | · K ) ≤ O ( n √ log n ) non-zero en tries. Graphs F and D hav e maxim um degree at most 2 K , and hence || N || ≤ 2 α | P | ∆ ( ||L ( F ) || + ||L ( D ) || ) ≤ 2 α n 1 − Θ( ε ) ∆ (2 K + 2 K ) ≤ α n 1 − Θ( ε ) . 10 4.3 Algorithm’s complexity W e start b y b ounding the complexity and width of the complete oracle. Lemma 14. L et ε ∈ [Θ(1 / log n ) , Θ(1)] . The or acle describ e d ab ove runs in O  n O ( ε ) · T maxflow ( n, m ) · log(max i w ( i )) · p olylog( n )  time and pr o duc es either a Θ(1) -b alanc e d vertex sep ar ator, or a valid fe e db ack matrix with sp e ctr al norm ρ = O  α √ log n n 1 − Θ( ε ) √ ε  (with high pr ob ability). Mor e over, every fe e db ack matrix has O ( m + n √ log n ) non-zer o entries. Pr o of. The algorithm for Theorem 8 can b e implemented in linear time (compute all norms and coun t), and it pro duces a matrix with sp ectral norm O  α n  . The n umber of non-zero entries is linear in n ≤ m . The remaining algorithm for the oracle is to rep eatedly run Algorithm 2, and p ossibly terminate early . Theorem 9 sho ws that if the algorithm does not pro duce a feedbac k matrix, then it produces a Θ(1)-balanced vertex separator. By Theorem 10, if it terminates early to pro duce a feedbac k matrix, then its feedbac k matrix has sp ectral norm at most β = Θ  6 α c ′ n ∆  = Θ  α √ log n n √ ε  , and it pro duces a feedback matrix with at most O ( m ) non-zero entries. Theorem 13 shows that if it completes, then it pro duces a feedback matrix with sp ectral norm at most α n 1 − Θ( ε ) , and it pro duces a feedbac k matrix with at most O ( n √ log n ) non-zero entries. Eac h run of Algorithm 2 uses one maxflow call, so the oracle uses O ( n O ( ε ) ) maxflo w calls in total (in exp ectation). Eac h call runs in T maxflow ( n, m ) time, assuming that all edge capacities are in tegers b ounded b y a p olynomial in n . The latter condition can be achiev ed as follows. Recall that w e are allow ed to use any v alue β ∈ [ β 0 , 2 β 0 ]. W e hav e β 0 = Θ( α/ ( n ∆)) and hence β 0 ≥ c ′′ /n for some constan t c ′′ ∈ (0 , 1 / 2), since α ≥ 1 and ∆ ≤ Θ(1). W e can no w set β = p/q where p = ⌈ 2 n c ′′ β 0 ⌉ and q = ⌊ 2 n c ′′ ⌋ . Clearly , both integers p and q are p olynomially b ounded in n . T o incorp orate the weigh ts w ( i ) (whic h may b e larger than a p olynomial in n ), we apply a scaling reduction [5, App endix B]. The oracle succeeds in finding a large violating generalized matc hing in O ( n Θ( ε ) ) iterations of the matching algorithm with probability at least 1 /n Θ( ε ) b y the rev erse Marko v inequality , so running this p ortion of the oracle O ( n Θ( ε ) log n ) times indep endently in lo ckstep gives us a v alid answ er in the stated time complexity with high probability . Finally , w e can prov e the main theorem. Pr o of of The or em 1. As discussed throughout the pap er, the result is prov en using matrix m ulti- plicativ e weigh t up date, given in Algorithm 1. Binary searching for the optimal ob jectiv e v alue α uses O (log( n max i w ( i ))) iterations. Theorem 6 gives us our correctness guarantee, assuming that the iterations run without issue. There are tw o sto chastic comp onents of eac h iteration: The dimension-reduced matrix exponentiation step pro duces its stated guaran tees with high proba- bilit y (see Theorem 7), and the oracle also pro duces its stated guarantees with high probability (see Theorem 14). By standard union b ounds, since the num b er of iterations is p olynomial, the en tire algorithm succeeds with high probability . It remains only to chec k time complexit y of the algorithm. As stated in Theorem 6, running the MMWU algorithm for a fixed α tak es T = ⌈ 4 n 2 ρ 2 ln n δ 2 ⌉ iterations, where δ = α / 2 and ρ is the width of the oracle’s feedbac k matrices. Theorem 14 b ounds the width of the feedbac k matrices b y ρ = O  α √ log n n 1 − Θ( ε ) √ ε  . W e substitute this in to the iteration coun t to get: T = O  n 2Θ( ε ) log 2 n ε  = O ( n Θ( ε ) p olylog( n )). The time complexit y of each iteration consists of the time to p erform the matrix exp onentiation plus the run time of the oracle. As stated in Theorem 7, the (appro ximate) matrix exp onen tiation step uses a T a ylor appro xi- mation, which is dominated by at most k matrix-vector pro ducts for some k ∈ Θ(max( η ρT , ln 1 τ )), 11 where the implicit matrix b eing used is precisely the sum of the feedbac k matrices pro duced so far. W e may c ho ose τ ∈ Θ(1), so k = Θ( η ρT ) suffices. The sum of the feedback matrices will hav e at most O ( T m + T n √ log n ) non-zero en tries, so eac h matrix-v ector pro duct will use at most that man y op erations. 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