Approximation Resistant Predicates From Pairwise Independence

Appro ximation Resistan t Prediates F rom P airwise Indep endene P er Austrin ∗ KTH  Ro y al Institute of T e hnology Sto  kholm, Sw eden El hanan Mossel † U.C. Berk eley USA 6 De 2007 Abstrat W e study the appro ximabilit y of prediates on k v ariables from a do- main [ q ] , and giv e a new suien t ondition for su h prediates to b e appro ximation resistan t under the Unique Games Conjeture. Sp ei- ally , w e sho w that a prediate P is appro ximation resistan t if there exists a balaned pairwise indep enden t distribution o v er [ q ] k whose supp ort is on tained in the set of satisfying assignmen ts to P . Using onstrutions of pairwise indep enen t distributions this result implies that • F or general k ≥ 3 and q ≥ 2 , the Max k -CSP q problem is UG-hard to appro ximate within q ⌈ log 2 k +1 ⌉− k + ǫ . • F or k ≥ 3 and q prime p o w er, the hardness ratio is impro v ed to kq ( q − 1) /q k + ǫ . • F or the sp eial ase of q = 2 , i.e., b o olean v ariables, w e an sharp en this b ound to ( k + O ( k 0 . 525 )) / 2 k + ǫ , impro ving up on the b est pre- vious b ound of 2 k/ 2 k + ǫ (Samoro dnitsky and T revisan, STOC'06) b y essen tially a fator 2 . • Finally , for q = 2 , assuming that the famous Hadamard Conjeture is true, this an b e impro v ed ev en further, and the O ( k 0 . 525 ) term an b e replaed b y the onstan t 4 . 1 In tro dution In the Max k -CSP problem, w e are giv en a set of onstrain ts o v er a set of b o olean v ariables, ea h onstrain t b eing a b o olean funtion ating on at most k of the v ariables. The ob jetiv e is to nd an assignmen t to the v ariables satisfying as man y of the onstrain ts as p ossible. This problem is NP-hard for an y k ≥ 2 , and as a onsequene, a lot of resear h has b een fo used on studying ho w w ell the problem an b e appro ximated. W e sa y that a (randomized) algorithm has ∗ E-mail: austrinkth.se . Resear h funded b y Sw edish Resear h Counil Pro jet Num b er 50394001. † E-mail: mosselstat.berkeley.edu . Resear h supp orted b y BSF gran t 2004105, NSF CAREER a w ard DMS 0548249 and DOD ONR gran t N0014-07-1-05-06 1 appr oximation r atio α if, for all instanes, the algorithm is guaran teed to nd an assignmen t whi h (in exp etation) satises at least α · Opt of the onstrain ts, where Opt is the maxim um n um b er of sim ultaneously satised onstrain ts, o v er an y assignmen t. A partiularly simple appro ximation algorithm is the algorithm whi h simply pi ks a random assignmen t to the v ariables. This algorithm has a ratio of 1 / 2 k . It w as rst impro v ed b y T revisan [22 ℄ who ga v e an algorithm with ratio 2 / 2 k for Max k -CSP . Reen tly , Hast [ 8℄ ga v e an algorithm with ratio Ω( k / (log k 2 k )) , whi h w as subsequen tly impro v ed b y Charik ar et al. [5 ℄ who ga v e an algorithm with appro ximation ratio c · k / 2 k , where c > 0 . 44 is an absolute onstan t. The PCP Theorem implies that the Max k -CSP problem is NP-hard to appro ximate within 1 /c k for some onstan t c > 1 . Samoro dnitsky and T re- visan [20 ℄ impro v ed this hardness to 2 2 √ k / 2 k , and this w as further impro v ed to 2 √ 2 k / 2 k b y Engebretsen and Holmerin [7℄. Finally , Samoro dnitsky and T revisan [21 ℄ pro v ed that, if the Unique Games Conjeture [12 ℄ is true, then the Max k -CSP problem is hard to appro ximate within 2 k / 2 k . T o b e more preise, the hardness they obtained w as 2 ⌈ log 2 k +1 ⌉ / 2 k , whi h is ( k + 1) / 2 k for k = 2 r − 1 , but an b e as large as 2 k / 2 k for general k . Th us, the urren t gap b et w een hardness and appro ximabilit y is a small onstan t fator of 2 / 0 . 44 . F or a prediate P : { 0 , 1 } k → { 0 , 1 } , the Max CSP ( P ) problem is the sp eial ase of Max k -CSP in whi h all onstrain ts are of the form P ( l 1 , . . . , l k ) , where ea h literal l i is either a v ariable or a negated v ariable. F or this problem, the random assignmen t algorithm a hiev es a ratio of m/ 2 k , where m is the n um b er of satisfying assignmen ts of P . Surprisingly , it turns out that for ertain  hoies of P , this is the b est p ossible algorithm. In a elebrated result, Håstad [10 ℄ sho w ed that for P ( x 1 , x 2 , x 3 ) = x 1 ⊕ x 2 ⊕ x 3 , the Max CSP (P) problem is hard to appro ximate within 1 / 2 + ǫ . Prediates P for whi h it is hard to appro ximate the Max CSP ( P ) problem b etter than a random assignmen t, are alled appr oximation r esistant . A sligh tly stronger notion is that of her e ditary appro ximation resistane  a prediate P is hereditary appro ximation resistan t if all prediates implied b y P are appro xima- tion resistan t. A natural and imp ortan t question is to understand the struture of appro ximation resistane. F or k = 2 and k = 3 , this question is resolv ed  prediates on 2 v ariables are nev er appro ximation resistan t, and a prediate on 3 v ariables is appro ximation resistan t if and only if it is implied b y an X OR of the three v ariables [10 , 23 ℄. F or k = 4 , Hast [9℄ managed to lassify most of the prediates with resp et to to appro ximation resistane, but for this ase there do es not app ear to b e as nie a  haraterization as there is in the ase k = 3 . It turns out that, assuming the Unique Games Conjeture, most prediates are in fat hereditary appro ximation resistan t  as k gro ws, the fration of su h pred- iates tend to 1 [11 ℄. Th us, instead of attempting to understand the seemingly ompliated struture of appro ximation resistan t prediates, one migh t try to understand the p ossibly easier struture of hereditary appro ximation resistan t prediates, as these onstitute the v ast ma jorit y of al l prediates. A natural approa h for obtaining strong inappro ximabilit y for the Max k - CSP problem is to sear h for appro ximation resistan t prediates with v ery few aepting inputs. This is indeed ho w all men tioned hardness results for Max k -CSP ome ab out (exept the one implied b y the PCP Theorem). It is natural to generalize the Max k -CSP problem to v ariables o v er a 2 domain of size q , rather than just b o olean v ariables. Without loss of generalit y w e ma y assume that the domain is [ q ] . W e all this the Max k -CSP q problem. F or Max k -CSP q , the random assignmen t giv es a 1 /q k -appro ximation, and an y f ( k ) -appro ximation algorithm for the Max k -CSP problem giv es a f ( k ⌈ log 2 q ⌉ ) - appro ximation algorithm for the Max k -CSP q problem. Th us, Charik ar et al.'s algorithm giv es a 0 . 44 k log 2 q /q k -appro ximation in the ase that q is a p o w er of 2 . The b est previous inappro ximabilit y for the Max k -CSP q problem is due to Engebretsen [6 ℄, who sho w ed that the problem is NP-hard to appro ximate within q O ( √ k ) /q k . Similarly to q = 2 , w e an dene the Max CSP (P) problem for P : [ q ] k → { 0 , 1 } . Here, there are sev eral natural w a ys of generalizing the notion of a literal. One p ossible denition is to sa y that a literal l is of the form π ( x i ) , for some v ariable x i and p erm utation π : [ q ] → [ q ] . A striter denition is to sa y that a literal is of the form x i + a , where, again, x i is a v ariable, and a ∈ [ q ] is some onstan t. In this pap er, w e use the seond, striter, denition. As this is a sp eial ase of the rst denition, our hardness results apply also to the rst denition. 1.1 Our on tributions Our main result is the follo wing: Theorem 1.1. L et P : [ q ] k → { 0 , 1 } b e a k -ary pr e di ate over [ q ] , and let µ b e a distribution over [ q ] k suh that Pr x ∈ ([ q ] k ,µ ) [ P ( x )] = 1 and for al l 1 ≤ i 6 = j ≤ k and al l a, b ∈ [ q ] , it holds that Pr x ∈ ([ q ] k ,µ ) [ x i = a, x j = b ] = 1 /q 2 . Then, for any ǫ > 0 , the UGC implies that the Max CSP ( P ) pr oblem is NP- har d to appr oximate within | P − 1 (1) | q k + ǫ, i.e., P is her e ditary appr oximation r esistant. Using onstrutions of pairwise indep enden t distributions, w e obtain the follo wing orollaries: Theorem 1.2. F or any k ≥ 3 , q ≥ 2 , and ǫ > 0 , it is UG-har d to appr oximate the Max k -CSP q pr oblem within q ⌈ log 2 k +1 ⌉ q k + ǫ < k log 2 q · q q k + ǫ. In the sp e ial  ase that k = 2 r − 1 for some r the har dness r atio impr oves to k log 2 q q k + ǫ. 3 This already onstitutes a signian t impro v emen t up on the q O ( √ k ) /q k - hardness of Engebretsen, and in the ase that q is a prime p o w er w e an impro v e this ev en further. Theorem 1.3. F or any k ≥ 3 , q = p e for some prime p , and ǫ > 0 , it is UG-har d to appr oximate the Max k -CSP q pr oblem within k ( q − 1 ) q q k + ǫ. In the sp e ial  ase that k = ( q r − 1) / ( q − 1) for some r , the har dness r atio impr oves to k ( q − 1 ) + 1 q k + ǫ ≤ k q q k + ǫ. Neither of these t w o theorems impro v e up on the results of [21 ℄ for the ase of q = 2 . Ho w ev er, the follo wing theorem do es. Theorem 1.4. F or any k ≥ 3 and ǫ > 0 , it is UG-har d to appr oximate the Max k -CSP pr oblem within k + O ( k 0 . 525 ) 2 k + ǫ. If the Hadamar d Conje tur e is true, it is UG-har d to appr oximate the Max k -CSP pr oblem within 4 ⌈ ( k + 1) / 4 ⌉ 2 k + ǫ ≤ k + 4 2 k + ǫ Th us, w e impro v e the hardness of [21 ℄ b y essen tially a fator 2 , dereasing the gap to the b est algorithm from roughly 2 / 0 . 44 to roughly 1 / 0 . 44 . 1.2 Related w ork It is in teresting to ompare our results to the results of Samordnitsky and T re- visan [21 ℄. Reall that using the Go w ers norm, [ 21 ℄ pro v e that the Max k -CSP problem has a hardness fator of 2 ⌈ log 2 k +1 ⌉ / 2 k , whi h is ( k + 1) / 2 k for k = 2 r − 1 , but an b e as large as 2 k / 2 k for general k . Our pro of uses the same v ersion of the UGC, but the analysis is more diret and more general. The pro of of [21 ℄ requires us to w ork sp eially with a lin- earit y h yp er-graph test for the long o des. F or this test, the suess probabilit y is sho wn to b e losely related to the Go w ers inner pro dut of the long o des. In partiular, in the soundness analysis it is sho wn that if the v alue of this test is to o large, it follo ws that the Go w ers norm is larger than for random funtions. F rom this it is sho wn that at least t w o of the funtions ha v e large inuenes whi h in turns allo ws us to obtain a go o d solution for the UGC. Our onstrution on the other hand allo ws an y pairwise distribution to dene a long-o de test. Using [16 ℄ w e sho w that if a olletion of supp osed long o des do es b etter than random for this long o de test, then at least t w o of them ha v e large inuenes. Our pro of has a n um b er of adv an tages: rst it applies to an y pairwise inde- p enden t distribution. This should b e ompared to [ 21 ℄ that require us to w ork 4 sp eially with the h yp er-graph linearit y test. In partiular our results allo w us to obtain hardness results for Max CSP ( P ) for a wide range of P 's. The results are general enough to aomo date an y domain [ q ] (it is not lear if the results of [21 ℄ extend to larger domains), and w e are also able to obtain a b etter hardness fator for most v alues of k ev en in the q = 2 ase. Also, our pro of uses b ounds on exp etations of pro duts under ertain t yp es of orrelation, putting it in the same general framew ork as man y other UGC- based hardness results, in partiular those for 2 -CSPs [ 13 , 14 , 2, 3 , 18 ℄. Finally , our pro of giv es parametrized hardness in the follo wing sense. W e giv e a family of hardness assumptions, alled the ( t, k ) -UGC. All of these as- sumptions follo w from the UGC, and in partiular the ase t = 2 is kno wn to b e equiv alen t to the UGC. Ho w ev er, the ( t, k ) -UGC assumption is w eak er for larger v alues of t . F or ea h v alue of t our results imply a dieren t hardness of appro ximation fator. Sp eially , if the ( t, k ) -UGC is true for some t ≥ 3 , then the Max k -CSP problem is NP-hard to appro ximate within O  k ⌈ t/ 2 ⌉− 1 / 2 k  . Th us, ev en the (4 , k ) -UGC giv es a hardness of O ( k / 2 k ) , and for t < √ k / log k , the ( t, k ) -UGC giv es a hardness b etter than the b est unonditional result kno wn [7℄. 2 Denitions 2.1 Unique Games W e use the follo wing form ulation of the Unique Lab el Co v er Problem: giv en is a k -uniform h yp ergraph, where for ea h edge ( v 1 , . . . , v k ) there are k p erm utations π 1 , . . . , π k on [ L ] . W e sa y that an edge ( v 1 , . . . , v k ) with p erm utations π 1 , . . . , π k is t -wise satised b y a lab elling ℓ : V → [ L ] if there are i 1 < i 2 < . . . < i t su h that π i 1 ( ℓ ( v i 1 )) = π i 2 ( ℓ ( v i 2 )) = . . . = π i t ( ℓ ( v i t )) . W e sa y that an edge is ompletely satised b y a lab elling if it is k -wise satised. W e denote b y Opt t ( X ) ∈ [0 , 1 ] the maxim um fration of t -wise satised edges, o v er an y lab elling. Note that Opt t +1 ( X ) ≤ Opt t ( X ) . The follo wing onjeture is kno wn to follo w from the Unique Games Conje- ture (see details b elo w). Conjeture 2.1. F or any 2 ≤ t ≤ k , and δ > 0 , ther e exists an L > 0 suh that it is NP-har d to distinguish b etwe en k -ary Unique L ab el Cover instan es X with lab el set [ L ] with Opt k ( X ) ≥ 1 − δ , and Opt t ( X ) ≤ δ . F or partiular v alues of t and k w e will refer to the orresp onding sp eial ase of the ab o v e onjeture as the ( t, k ) - Unique Games Conje tur e (or the ( t, k ) -UGC). Khot's original form ulation of the Unique Games Conjeture [12 ℄ is then exatly the (2 , 2) -UGC, and Khot and Regev [15 ℄ pro v ed that this onjeture is equiv alen t to the (2 , k ) -UGC for all k , whi h is what Samoro dnitsky and T revisan [21 ℄ used to obtain hardness for Max k -CSP . In this pap er, w e mainly use the (3 , k ) -UGC to obtain our hardness results. Clearly , sine Opt t +1 ( X ) ≤ Opt t ( X ) , the ( t, k ) -UGC implies the ( t + 1 , k ) -UGC, so our assumption is implied b y the Unique Games Conjeture. But whether the on v erse holds, or whether there is hop e of pro ving this onjeture (or, sa y , 5 the ( k , k ) -UGC for large k ) without pro ving the Unique Games Conjeture, is not lear, and should b e an in teresting diretion for future resear h. 2.2 Inuenes It is w ell kno wn (see e.g. [13 ℄) that ea h funtion f : [ q ] n → R admits a unique Efr on-Stein de  omp osition: f = P S ⊆ [ n ] f S where • The funtion f S dep ends on x S = ( x i : i ∈ S ) only . • F or ev ery S ′ 6⊆ S , and ev ery y S ′ ∈ [ q ] S ′ it holds that E [ f S ( x S ) | x S ′ = y S ′ ] = 0 . F or m ≤ n w e write f ≤ m = P S : | S |≤ m f S for the m -degree expansion of f . W e no w dene the inuen e of the i th  o or dinate on f , denoted b y Inf i ( f ) b y Inf i ( f ) = E x [V ar x i [ f ( x )]] . (1) W e dene the m -de gr e e inuen e of the i th  o or dinate on f , denoted b y Inf ≤ m i ( f ) b y Inf i ( f ≤ m ) . Reall that the inuene Inf i ( f ) measures ho w m u h the funtion f dep ends on the i 'th v ariable, while the lo w degree inuenes Inf ≤ m i ( f ) measures this for the lo w part of the expansion of f . The later quan tit y is losely related to the inuene of f on sligh tly noisy inputs. An imp ortan t prop ert y of lo w-degree inuenes is that n X i =1 Inf ≤ m i ( f ) ≤ m V ar[ f ] , implying that the n um b er of o ordinates with large lo w-degree inuene m ust b e small. In partiular, if f : [ q ] n → [0 , 1] , then the the n um b er of o ordinates with lo w-degree inuene at least τ is at most τ /m . 2.3 Correlated Probabilit y Spaes W e will b e in terested in probabilit y distributions supp orted in P − 1 (1) ⊆ [ q ] k . It w ould b e useful to follo w [ 16 ℄ and view [ q ] k with su h probabilit y measure as a olletion of k  orr elate d sp a es orresp onding to the k o ordinates. W e pro eed with formal denitions of t w o and k orrelated spaes. Denition 2.2. Let (Ω , µ ) b e a probabilit y spae o v er a nite pro dut spae Ω = Ω 1 × Ω 2 . The  orr elation b et w een Ω 1 and Ω 2 (with resp et to µ ) is ρ (Ω 1 , Ω 2 ; µ ) = sup { Cov[ f 1 ( x 1 ) f 2 ( x 2 )] : f i : Ω i → R , V ar[ f i ( x i )] = 1 } , where ( x 1 , x 2 ) is dra wn from (Ω , µ ) . Denition 2.3. Let (Ω , µ ) b e a probabilit y spae o v er a nite pro dut spae Q k i =1 Ω i , and let Ω S = Q i ∈ S Ω i . The orrelation of Ω 1 , . . . , Ω k (with resp et to µ ) is ρ (Ω 1 , . . . , Ω k ; µ ) = max 1 ≤ i ≤ k − 1 ρ (Ω { 1 ,...,i } , Ω { i +1 ,...,k } ; µ ) 6 Of partiular in terest to us is the ase where orrelated spaes are dened b y a measure that it t -wise indep enden t. Denition 2.4. Let (Ω , µ ) b e a probabilit y spae o v er a pro dut spae Ω = Q k i =1 Ω i . W e sa y that µ is t -wise indep enden t if, for an y  hoie of i 1 < i 2 < . . . < i t and b 1 , . . . , b t with b j ∈ Ω i j , w e ha v e that Pr w ∈ (Ω ,µ ) [ w i 1 = b 1 , . . . , w i s = b s ] = t Y j =1 Pr w ∈ (Ω ,µ ) [ w i j = b j ] W e sa y that (Ω , µ ) is b alan e d if for ev ery i ∈ [ k ] , b ∈ Ω i , w e ha v e that Pr w ∈ (Ω ,µ ) [ w i = b ] = 1 / | Ω i | . The follo wing theorem onsiders lo w inuene funtions that at on orre- lated spaes where the orrelation is giv en b y a t -wise indep enden t probabilit y measure for t ≥ 2 . It sho ws that in this ase, the funtions ha v e almost the same distribution as if they w ere ompletely indep enden t. Moreo v er, the result holds ev en if some of the funtions ha v e large inuenes as long as in ea h o ordinate not more than t funtions ha v e large inuenes. Theorem 2.5 ([16 ℄, Theorem 6.6 and Lemma 6.9) . L et (Ω , µ ) b e a nite pr ob- ability sp a e over Ω = Q k i =1 Ω i with the fol lowing pr op erties: (a) µ is t -wise indep endent. (b) F or al l i ∈ [ k ] and b i ∈ Ω i , µ i ( b i ) > 0 . () ρ (Ω 1 , . . . , Ω k ; µ ) < 1 . Then for al l ǫ > 0 ther e exists a τ > 0 and d > 0 suh that the fol lowing holds. L et f 1 , . . . , f k b e funtions f i : Ω n i → [0 , 1] satisfying that, for al l 1 ≤ j ≤ n , |{ i : Inf ≤ d j ( f i ) ≥ τ }| ≤ t. Then      E w 1 ,...,w n " k Y i =1 f i ( w 1 ,i , . . . , w n,i ) # − k Y i =1 E w 1 ,...,w n [ f i ( w 1 ,i , . . . , w n,i )]      ≤ ǫ, wher e w 1 , . . . , w n ar e dr awn indep endently fr om (Ω , µ ) , and w i,j ∈ Ω j denotes the j th  o or dinate of w i . Note that a suien t ondition for () to hold in the ab o v e theorem is that for all w ∈ Ω , µ ( w ) > 0 . Roughly sp eaking, the basi idea b ehind the theorem and its pro of is that lo w inuene funtions annot detet dep endenies of high order  in partiular if the underlying measure is pairwise indep enden t, then lo w inuene funtions of dieren t o ordinates are essen tially indep enden t. 7 3 Main theorem In this setion, w e pro v e our main theorem. Note that it is a generalization of Theorem 1.1 . Theorem 3.1. L et P : [ q ] k → { 0 , 1 } b e a k -ary pr e di ate over a (nite) domain of size q , and let µ b e a b alan e d t -wise indep endent distribution over [ q ] k suh that Pr x ∈ ([ q ] k ,µ ) [ P ( x )] > 0 . Then, for any ǫ > 0 , the ( t + 1 , k ) -UGC implies that the Max CSP ( P ) pr oblem is NP-har d to appr oximate within | P − 1 (1) | q k · P r x ∈ ([ q ] k ,µ ) [ P ( x )] + ǫ In partiular, note that if Pr x ∈ ([ q ] k ,µ ) [ P ( x )] = 1 , i.e., if the supp ort of µ is en tirely on tained in the set of satisfying assignmen ts to P , then P is appro xi- mation resistan t. It is also hereditary appro ximation resistan t, sine the supp ort of µ will still b e on tained in P − 1 (1) when w e add more satisfying assignmen ts to P . Redution. Giv en a k -ary Unique Lab el Co v er instane X , the pro v er writes do wn the table of a funtion f v : [ q ] L → [ q ] for ea h v , whi h is supp osed to b e the long o de of the lab el of the v ertex v . F urthermore, w e will assume that f v is folded, i.e., that for ev ery x ∈ [ q ] k and a ∈ [ q ] , w e ha v e f v ( x + ( a, . . . , a )) = f v ( x ) + a (where the denition of  +  in [ q ] is arbitrary as long as ([ q ] , +) is an Ab elian group). When reading the v alue of f v ( x 1 , . . . , x L ) , the v erier an enfore this ondition b y instead querying f v ( x 1 − x 1 , x 2 − x 1 , . . . , x L − x 1 ) and adding x 1 to the result. Let η > 0 b e a parameter, the v alue of whi h will b e determined later, and dene a probabilit y distribution µ ′ on [ q ] k b y µ ′ ( w ) = (1 − η ) · µ ( w ) + η · µ U ( w ) , where µ U is the uniform distribution on [ q ] k , i.e., µ U ( w ) = 1 /q k . Giv en a pro of Σ = { f v } v ∈ V of supp osed long o des for a go o d lab elling of X , the v erier  he ks Σ as follo ws. Algorithm 1: The v erier V V ( X , Σ = { f v } v ∈ V ) (1) Pi k a random edge e = ( v 1 , . . . , v k ) with p erm utations π 1 , . . . , π k . (2) F or ea h i ∈ [ L ] , dra w w i randomly from ([ q ] k , µ ′ ) . (3) F or ea h j ∈ [ k ] , let x j = w 1 ,j . . . w L,j , and let b j = f v j π j ( x j ) . (4) A ept if P ( b 1 , . . . , b k ) . Lemma 3.2 (Completeness) . F or any δ , if Opt k ( X ) ≥ 1 − δ , then ther e is a pr o of Σ suh that Pr[ V ( X , Σ) a  epts ] ≥ (1 − δ )(1 − η ) Pr w ∈ ([ q ] k ,µ ) [ P ( w )] 8 Pr o of. T ak e a lab elling ℓ for X su h that a fration ≥ 1 − δ of the edges are k -wise satised, and let f v : [ q ] L → [ q ] b e the long o de of the lab el ℓ ( v ) of v ertex v . Let ( v 1 , . . . , v k ) b e an edge that is k -wise satised b y ℓ . Then f v 1 π 1 = f v 2 π 2 = . . . = f v k π k , ea h b eing the long o de of i := π 1 ( ℓ ( v 1 )) . The probabilit y that V aepts is then exatly the probabilit y that P ( w i ) is true, whi h, sine w i is dra wn from ([ q ] k , µ ) with probabilit y 1 − η , is at least (1 − η ) Pr w ∈ ([ q ] k ,µ ) [ P ( w )] . The probabilit y that the edge e  hosen b y the v erier in step 1 is satised b y ℓ is at least 1 − δ , and so w e end up with the desired inequalit y . Lemma 3.3 (Soundness) . F or any ǫ > 0 , η > 0 , ther e is a  onstant δ := δ ( ǫ, η , t, k, q ) > 0 , suh that if Opt t +1 ( X ) < δ , then for any pr o of Σ , we have Pr[ V ( X , Σ) a  epts ] ≤ | P − 1 (1) | q k + ǫ Pr o of. Assume that Pr[ V ( X , Σ) aepts ] > | P − 1 (1) | q k + ǫ. (2) W e need to pro v e that this implies that there is a δ := δ ( ǫ, η , t, k, q ) > 0 su h that Opt t +1 ( X ) ≥ δ . Equation 2 implies that for a fration of at least ǫ/ 2 of the edges e , the probabilit y that V ( X , Σ) aepts when  ho osing e is at least | P − 1 (1) | q k + ǫ/ 2 . Let e = ( v 1 , . . . , v k ) with p erm utations π 1 , . . . , π k b e su h a go o d edge. F or v ∈ V and a ∈ [ q ] , dene g v, a : [ q ] L → { 0 , 1 } b y g v, a ( x ) =  1 if f v ( x ) = a 0 otherwise . The probabilit y that V aepts when  ho osing e is then exatly X x ∈ P − 1 (1) E w 1 ,...,w L " k Y i =1 g v i ,x i π i ( w 1 ,i , . . . , w L,i ) # , whi h, b y the  hoie of e , is greater than | P − 1 (1) | /q k + ǫ/ 2 . This implies that there is some x ∈ P − 1 (1) su h that E w 1 ,...,w L " k Y i =1 g v i ,x i π i ( w 1 ,i , . . . , w L,i ) # > 1 /q k + ǫ ′ = k Y i =1 E w 1 ,...,w L [ g v i ,x i π i ( w 1 ,i , . . . , w L,i )] + ǫ ′ , where ǫ ′ = ǫ/ 2 / | P − 1 (1) | and the last equalit y uses that, b eause f v i is folded and µ is balaned, w e ha v e E w 1 ,...,w L [ g v i ,x i ( w 1 ,i , . . . , w L,i )] = 1 /q . Note that b eause b oth µ and µ U are t -wise indep enden t, µ ′ is also t -wise indep enden t. Also, w e ha v e that for ea h w ∈ [ q ] k , µ ′ ( w ) ≥ η /q k > 0 , whi h implies b oth onditions (b) and () of Theorem 2.5 . Then, the on trap ositiv e form ulation of Theorem 2.5 implies that there is an i ∈ [ L ] and at least t + 1 9 indies J ⊆ [ k ] su h that Inf ≤ d π − 1 j ( i ) ( g v j ,x j ) = Inf ≤ d i ( g v j ,x j π j ) ≥ τ for all j ∈ J , where τ and d are funtions of ǫ , η , t , k , and q . The pro ess of onstruting a go o d lab elling of X from this p oin t is standard. F or ompleteness, w e giv e a pro of in the app endix. Sp eially , Lemma A.1 giv es that Opt t +1 ( X ) ≥ ǫ/ 2  τ d · q  t +1 , whi h is a funtion of ǫ , η , t , k , and q , as desired. It is no w straigh tforw ard to pro v e Theorem 3.1 . Pr o of of The or em 3.1 . Let c = Pr x ∈ ([ q ] k ,µ ) [ P ( x )] , s = | P − 1 (1) | /q k and η = min(1 / 4 , ǫc 4 s ) . Note that sine the statemen t of the Theorem requires c > 0 w e also ha v e s > 0 and η > 0 . Assume that the ( t + 1 , k ) -UGC is true, and pi k L large enough so that it is NP-hard to distinguish b et w een k -ary Unique Lab el Co v er instanes X with Opt t +1 ( X ) ≤ δ and Opt k ( X ) ≥ 1 − δ , where δ = min( η , δ ( ǫc/ 4 , η , t, k, q )) , where δ ( . . . ) is the funtion from Lemma 3.3 . By Lemmas 3.2 and 3.3 , w e then get that it is NP-hard to distinguish b et w een Max CSP ( P ) instanes with Opt ≥ (1 − δ )(1 − η ) c ≥ (1 − 2 η ) c and Opt ≤ s + ǫc/ 4 . In other w ords, it is NP-hard to appro ximate the Max CSP ( P ) problem within a fator s + ǫc/ 4 (1 − 2 η ) c ≤ s (1 + 4 η ) c + (1 + 4 η ) ǫ/ 4 ≤ s/c + ǫ 4 Inappro ximabilit y for Max k -CSP q As a simple orollary to Theorem 3.1 , w e ha v e: Corollary 4.1. L et t ≥ 2 and let µ b e a b alan e d t -wise indep endent distribution over [ q ] k . Then the ( t + 1 , k ) -UGC implies that that Max k -CSP q pr oblem is NP-har d to appr oximate within | Supp( µ ) | q k Th us, w e ha v e redued the problem of obtaining strong inappro ximabilit y for Max k -CSP q to the problem of nding small t -wise indep enden t distributions. As w e are mainly in terested in the strongest p ossible results that an b e obtained b y this metho d, our main fo us will b e on pairwise indep endene, i.e, t = 2 . Ho w ev er, let us rst men tion t w o simple orollaries for general v alues of t . F or q = 2 , it is w ell-kno wn that the binary BCH o de giv es a t -wise indep en- den t distribution o v er { 0 , 1 } k with supp ort size O ( k ⌊ t/ 2 ⌋ ) [1 ℄. In other w ords, the ( t + 1 , k ) -UGC implies that the Max k -CSP problem is NP-hard to ap- pro ximate within O ( k ⌈ t/ 2 ⌉ / 2 k ) . Note in partiular that the (4 , k ) -UGC sues to get a hardness of O ( k / 2 k ) for Max k -CSP , whi h is tigh t up to a onstan t fator. F or q a prime p o w er and large enough so that q ≥ k , there are t -wise in- dep enden t distributions o v er [ q ] k with supp ort size q t based on ev aluating a random degree- t p olynomial o v er F q . Th us, in this setting, the ( t + 1 , k ) -UGC implies a hardness fator of q t − k for the Max k -CSP q problem. 10 In the remainder of this setion, w e will fo us on the details of onstrutions of pairwise indep endene, giving hardness for Max k -CSP q under the (3 , k ) - UGC. 4.1 Theorems 1.2 and 1.3 The pairwise indep enden t distributions used to giv e Theorems 1.2 and 1.3 are b oth based on the follo wing simple lemma, whi h is w ell-kno wn but stated here in a sligh tly more general form than usual: Lemma 4.2. L et R b e a nite  ommutative ring, and let u, v ∈ R n b e two ve tors over R suh that u i v j − u j v i ∈ R ∗ for some i, j . 1 L et X ∈ R n b e a uniformly r andom ve tor over R n and let µ b e the pr ob ability distribution over R 2 of ( h u, X i , h v , X i ) ∈ R 2 . Then µ is a b alan e d p airwise indep endent distribution. Pr o of. Without loss of generalit y , assume that i = 1 and j = 2 . It sues to pro v e that, for all ( a, b ) ∈ R 2 and an y  hoie of v alues of X 3 , . . . , X n , w e ha v e Pr[( h u, X i , h v , X i ) = ( a, b ) | X 3 , . . . , X n ] = 1 / | R | 2 . F or this to b e true, w e need that the system  u 1 X 1 + u 2 X 2 = a ′ v 1 X 1 + v 2 X 2 = b ′ has exatly one solution, where a ′ = a − P n i =3 u i X i and similarly for b ′ . This in turn follo ws diretly from the ondition on u and v . Consequen tly , giv en a set of m v etors in R n su h that an y pair of them satisfy the ondition of Lemma 4.2 , w e an onstrut a pairwise indep enden t distribution o v er R m with supp ort size | R | n . Let us no w pro v e Theorem 1.2 . Pr o of of The or em 1.2 . Let r = ⌈ log 2 k + 1 ⌉ . F or a nonempt y S ⊆ [ r ] , let u S ∈ Z r q b e the  harateristi v etor of S , i.e., u S,i = 1 if i ∈ S , and 0 otherwise. Then, for an y S 6 = T , the v etors u S and u T satisfy the ondition of Lemma 4.2, and th us, w e ha v e that ( h u S , X i ) S ⊆ [ r ] for a uniformly random X ∈ Z r q indues a balaned pairwise indep enden t distribution o v er Z 2 r − 1 q , with supp ort size q r . When k = 2 r − 1 w e get a hardness of q log 2 ( k ) − k , but for general v alues of k , in partiular k = 2 r − 1 , w e ma y lose up to a fator q . W e remark that for q = 2 this onstrution giv es exatly the prediate used b y Samoro dnitsky and T revisan [21℄, giving an inappro ximabilit y of 2 k / 2 k for all k , and ( k + 1) / 2 k for all k of the form 2 l − 1 . In tuitiv ely , it should b e lear that when w e ha v e more struture on R in Lemma 4.2 , w e should b e able to nd a larger olletion of v etors where ev ery pair satises the indep endene ondition. This in tuition leads us to Theo- rem 1.3 , dealing with the sp eial ase of Theorem 1.2 in whi h q is a prime p o w er. The onstrution of Theorem 1.3 is essen tially the same as that of [ 17 ℄. 1 R ∗ denotes the set of units of R . In the ase that R is a eld, the ondition is equiv alen t to sa ying that u and v are linearly indep enden t. 11 Pr o of of The or em 1.3 . Let r = ⌈ log q ( k ( q − 1) + 1) ⌉ , and n = ( q r − 1) / ( q − 1) ≥ k . Let P ( F r q ) b e the pro jetiv e spae o v er F r q , i.e., P ( F r q ) = ( F r q \ 0) / ∼ . Here ∼ is the equiv alene relation dened b y ( x 1 , . . . , x r ) ∼ ( y 1 , . . . , y r ) if there exists a c ∈ F ∗ q su h that x i = cy i for all i , i.e., if ( x 1 , . . . , x r ) and ( y 1 , . . . , y r ) are linearly indep enden t. W e then ha v e | P ( F r q ) | = ( q r − 1) / ( q − 1) = n . Cho ose n v etors u 1 , . . . , u n ∈ F r q as represen tativ es from ea h of the equiv a- lene lasses of P ( F r q ) . Then an y pair u i , u j satisfy the ondition of Lemma 4.2 , and as in Theorem 1.2 , w e get a balaned pairwise indep enden t distribution o v er F n q , with supp ort size q r . When k = ( q r − 1) / ( q − 1) , this giv es a hardness of k ( q − 1) + 1 , and for general k , in partiular k = ( q r − 1 − 1) / ( q − 1) + 1 , w e lose a fator q in the hardness ratio. Again, for q = 2 , this onstrution giv es the same prediate used b y Samoro d- nitsky and T revisan. In the ase that q ≥ k , w e get a hardness of q 2 /q k , the same fator as w e get from the general onstrution for t -wise indep endene men tioned at the b eginning of this setion. 4.2 Theorem 1.4 Let us no w lo ok loser at the sp eial ase of b o olean v ariables, i.e., q = 2 . So far, w e ha v e only giv en a dieren t pro of of Samoro dnitsky and T revisan's result, but w e will no w sho w ho w to impro v e this. An Hadamard matrix is an n × n matrix o v er ± 1 su h that H H T = nI , i.e., ea h pair of ro ws, and ea h pair of olumns, are orthogonal. Let h ( n ) denote the smallest n ′ ≥ n su h that there exists an n ′ × n ′ Hadamard matrix. It is a w ell-kno wn fat that Hadamard matries giv e small pairwise indep enden t distri- butions and th us giv e hardness of appro ximating Max k -CSP . T o b e sp ei, w e ha v e the follo wing prop osition: Prop osition 4.3. F or every k ≥ 3 , the (3 , k ) -UGC implies that the Max k - CSP pr oblem is UG-har d to appr oximate within h ( k + 1 ) / 2 k + ǫ . Pr o of. Let n = h ( k + 1) and let A b e an n × n Hadamard matrix, normalized so that one olumn on tains only ones. Remo v e n − k of the olumns, inluding the all-ones olumn, and let A ′ b e the resulting n × k matrix. Let µ : {− 1 , 1 } k → [0 , 1] b e the probabilit y distribution whi h assigns probabilit y 1 /n to ea h ro w of A ′ . Then µ is a balaned pairwise indep enden t distribution with | Supp( µ ) | = h ( k + 1 ) . It is w ell kno wn that Hadamard matries an only exist for n = 1 , n = 2 , and n ≡ 0 (mo d 4) . The famous Hadamar d Conje tur e asserts that Hadamard matries exist for all n whi h are divisible b y 4 , in other w ords, that h ( n ) = 4 ⌈ n/ 4 ⌉ ≤ n + 3 . It is also p ossible to get useful unonditional b ounds on h ( n ) . W e no w giv e one su h easy b ound. Theorem 4.4 ([19 ℄) . F or every o dd prime p and inte gers e, f ≥ 0 , ther e exists an n × n Hadamar d matrix H n wher e n = 2 e ( p f + 1) , whenever this numb er is divisible by 4 . Theorem 4.5 ([4℄) . Ther e exists an inte ger n 0 suh that for every n ≥ n 0 , ther e is a prime p b etwe en n and n + n 0 . 525 . 12 Corollary 4.6. W e have: h ( n ) ≤ n + O ( n 0 . 525 ) . Pr o of. Let p b e the smallest prime larger than n/ 2 , and let n ′ = 2( p + 1) ≥ n . Then, Theorem 4.4 asserts that there exists an n ′ × n ′ Hadamard matrix, so h ( n ) ≤ n ′ . If n is suien tly large ( n ≥ 2 n 0 ), then b y Theorem 4.5 , p ≤ n/ 2 + ( n/ 2) 0 . 525 and n ′ ≤ n + 2 n 0 . 525 , as desired. Theorem 1.4 follo ws from Prop osition 4.3 and Corollary 4.6 . It is probably p ossible to get a stronger unonditional b ound on h ( n ) than the one giv en b y Corollary 4.6 , b y using stronger onstrution te hniques than the one of Theorem 4.4 . 5 Disussion W e ha v e giv en a strong suien t ondition for prediates to b e hereditary ap- pro ximation resistan t under (a w eak ened v ersion of ) the Unique Games Con- jeture: it sues for the set of satisfying assignmen ts to on tain a balaned pairwise indep enden t distribution. Using onstrutions of small su h distribu- tions, w e w ere then able to onstrut appro ximation resistan t prediates with few aepting inputs, whi h in turn ga v e impro v ed hardness for the Max k - CSP q problem. There are sev eral asp ets here where there is ro om for in teresting further w ork: As men tioned earlier, w e do not kno w whether the ( t, k ) -UGC implies the standard UGC for large v alues of t . In partiular, pro ving the ( t, k ) -UGC for some t < √ k / log k w ould giv e hardness for Max k -CSP b etter than the b est urren t NP-hardness result. But ev en understanding the ( k , k ) -UGC seems lik e an in teresting question. A v ery natural and in teresting question is whether our ondition is also ne- essary for a prediate to b e hereditary appro ximation resistan t, i.e., if pairwise indep endene giv es a omplete  haraterization of hereditary appro ximation re- sistane. Finally , it is natural to ask whether our results for Max k -CSP q an b e pushed a bit further, or whether they are tigh t. F or the ase of b o olean v ariables, Hast [9 ℄ pro v ed that an y prediate aepting at most 2 ⌊ k / 2 ⌋ + 1 inputs is not appro ximation resistan t. F or k ≡ 2 , 3 (mod 4) this exatly mat hes the result w e get under the UGC and the Hadamard Conjeture (whi h for k = 2 r − 1 and k = 2 r − 2 is the same hardness as [ 21 ℄). F or k ≡ 0 , 1 (mo d 4) , w e get a gap of 2 b et w een ho w few satisfying assignmen ts an appro ximation resistan t prediate an and annot ha v e. Th us, the hitherto v ery suesful approa h of obtaining hardness for Max k - CSP b y nding small appro ximation resistan t prediate, an not b e tak en fur- ther, but there is still a small onstan t gap of roughly 1 / 0 . 44 to the b est urren t algorithm. It w ould b e in teresting to kno w whether the algorithm an b e im- pro v ed, or whether the hardest instanes of Max k -CSP are not Max CSP ( P ) instanes for some appro ximation resistan t P . F or larger q , this situation b eomes a lot w orse. When q = 2 l and k = ( q r − 1) / ( q − 1) , w e ha v e a gap of Θ( q / log 2 q ) b et w een the b est algorithm and the b est inappro ximabilit y , and for general v alues of q and k , the gap is ev en larger. 13 Referenes [1℄ Noga Alon, László Babai, and Alon Itai. 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More Eien t Queries in PCPs for NP and Impro v ed Appro ximation Hardness of Maxim um CSP. In Symp o- sium on The or eti al Asp e ts of Computer Sien e (ST A CS) , pages 194205, 2005. [8℄ Gusta v Hast. Appro ximating Max kCSP  Outp erforming a Random As- signmen t with Almost a Linear F ator. In ICALP 2005 , pages 956968, 2005. [9℄ Gusta v Hast. Be ating a R andom Assignment  Appr oximating Constr aint Satisfation Pr oblems . PhD thesis, KTH  Ro y al Institute of T e hnology , 2005. [10℄ Johan Håstad. Some optimal inappro ximabilit y results. Journal of the A CM , 48(4):798859, 2001. [11℄ Johan Håstad. On the appro ximation resistane of a random prediate. T o app ear in RANDOM-APPR O X, 2007. [12℄ Subhash Khot. On the p o w er of unique 2-pro v er 1-round games. In STOC 2002 , pages 767775, 2002. [13℄ Subhash Khot, Guy Kindler, El hanan Mossel, and Ry an O'Donnell. Opti- mal inappro ximabilit y results for max-ut and other 2-v ariable sps? 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Go w ers uniformit y , inuene of v ariables, and PCPs. In STOC 2006 , pages 1120, 2006. [22℄ Lua T revisan. P arallel Appro ximation Algorithms b y P ositiv e Linear Pro- gramming. A lgorithmi a , 21:7288, 1998. [23℄ Uri Zwi k. Appro ximation Algorithms for Constrain t Satisfation Problems In v olving at Most Three V ariables P er Constrain t. In SOD A 1998 , 1998. A Go o d lab ellings from inuen tial v ariables Lemma A.1. L et X b e a k -ary Unique L ab el Cover instan e. F urthermor e, for e ah vertex v , let f v : [ q ] k → [ q ] and dene g v, a ( x ) =  1 if f v i = a 0 otherwise . Then if ther e is a fr ation of at le ast ǫ e dges e = ( v 1 , . . . , v k ) with a ve tor a ∈ [ q ] k , an index i ∈ [ L ] and a set J ⊆ [ k ] of | J | = t indi es suh that Inf ≤ d π − 1 j ( i ) ( g v j ,a j ) ≥ τ (3) for al l j ∈ J , then Opt t ( X ) ≥ δ := ǫ  τ d · q  t . Pr o of. F or ea h v ∈ V , let C ( v ) = { i | Inf ≤ d i ( g v, a ) ≥ τ for some a ∈ [ q ] } . Note that | C ( v ) | ≤ q · d/τ . Dene a lab elling ℓ : V → [ L ] b y pi king, for ea h v ∈ V , a lab el ℓ ( v ) uniformly at random from C ( v ) (or an arbitrary lab el in ase C ( v ) is empt y). Let e = ( v 1 , . . . , v k ) b e an edge satisfying Equation 3 . Then for all j ∈ J , π − 1 j ( i ) ∈ C ( v j ) , and th us, the probabilit y that π j ( ℓ ( v j )) = i is 1 / | C ( v j ) | . This implies that the probabilit y that this edge is t -wise satised is at least Q j ∈ J 1 / | C ( v j ) | ≥  τ d · q  t . Ov erall, the total exp eted n um b er of edges that are t -wise satised b y ℓ is at least δ = ǫ  τ d · q  t , and th us Opt t ( X ) ≥ δ . 15