How to Play Unique Games on Expanders

Ho w to Pla y Unique Games o n Expanders Konstan tin Mak aryc hev ∗ Y ury Mak aryc hev † Abstract In this note we improv e a recent result b y Aro r a, Khot, K o lla, Steurer, T ulsiani, and Vishnoi on solving the Unique Games problem o n expanders. Given a (1 − ε )- s atisfiable ins tance of Unique Ga mes w ith the c onstraint graph G , our algorithm finds an assignment satisfying at least a 1 − C ε/h G fraction o f all co nstraints if ε < cλ G where h G is the edge expansion o f G , λ G is the seco nd smallest eigenv alue of the Laplacian of G , and C a nd c are some abs olute co ns tants. W e refer the reader to [1, 2, 3, 4, 7] for the motiv ation and a n ov erview of related work. 1 Preliminaries: Expanders, Unique Games and SDP 1.1 Unique Games and Expanders In t h is n ote we stud y the Unique Games problem on regular expand er s . Definition 1.1 (Unique Games Problem) . Given a c onstr aint g r aph G = ( V , E ) and a set of p ermutat ions π uv on the set [ k ] = { 1 , . . . , k } (for al l e dges ( u, v ) ), the go a l is t o assign a value (state) x u fr om [ k ] to e ach ve rtex u so as to s atisfy t he maximum numb er of c onstr aints o f th e for m π uv ( x u ) = x v . The c o st of a solution is the fr action of satisfie d c on str aints. W e assu me that the u n derlying graph G = ( V , E ) is a d -regular expander. Th e t wo key pa- rameters of the expan d er G a re the edge expansion h G and the seco n d eige nv al u e of the Laplacian λ G . The edge expansion g ives a lo w er b ound on the size o f ev ery cut: for ev ery subset of v ertices X ⊂ V , the size of the cut b et ween X and | V \ X | is at least h G × min( | X | , | V \ X | ) | V | | E | . It is formally defined as follo ws: h G = m in X ⊂ V  | δ ( X, V \ X ) | | E |  min( | X | , | V \ X | ) | V |  , here δ ( X , V \ X ) d enotes the c u t — t h e set of edges going from X to V \ X . One can think of the second eigen v alue of the Laplacian L G ( u, v ) =      1 , if u = v − 1 /d, if ( u, v ) ∈ E 0 , otherwise. ∗ IBM T.J. W atson Research Center, Y orktown Heights, NY 10598. † Microsof t R esearc h N ew England, One Memoria l Drive, Cambridge, MA 02142. 1 as of contin uous r elaxation of the ed ge expansion. Not e that the smallest eige nv al u e of L G is 0; and t h e corresp onding eigen ve ctor is a ve ctor of all 1’s, denote d by 1 . Thus λ G = min x ⊥ 1 h x, L G x i k x k 2 . Cheeger’s inequalit y , h 2 G / 8 ≤ λ G ≤ h G , sho ws that h G and λ G are cl osely related; h o wev er λ G can b e m uch smaller than h G (the lo w er b ound in th e inequalit y is tight). 1.2 Results of Arora, Khot, Kolla, Steurer, T ulsiani, and Vishnoi In a recent w ork [1], Arora, Khot, Kolla, Steurer, T ulsiani, and Vishnoi show ed ho w giv en a (1 − ε ) satisfiable instance of Unique Games (i. e. an instance in whic h the optimal solution sat isfi es at least a (1 − ε ) fr action of constraints), one can obtain a solution of cost 1 − C ε λ G log  λ G ε  in p olynomial time, her e C is an absolute constan t. W e imp ro ve their result an d s ho w that, if the ratio ε/λ G is less th an some u niv ersal p o sitive constant c , one can obtain a solution of cost 1 − C ′ ε h G in p olynomial time. As mentio n ed ab ov e, λ G can b e significan tly smaller than h G , then our resu lt giv es m u c h b etter appro ximation guarante e. H ow ev er, ev en if λ G ≈ h G , our b oun d is asymptotically stronger, since 1 − C ′ ε h G ≥ 1 − C ′ ε λ G (our b ound do es n ot h av e a log ( λ G /ε ) f actor). It is an int eresting op en question, if one can replace the condition ε/λ G < c with ε/h G < c . 1.3 Semidefinite Relaxa t ion W e use the standard SDP relaxation for the Unique Games problem. minimize 1 2 | E | X ( u,v ) ∈ E k X i =1 k u i − v π uv ( i ) k 2 sub ject to ∀ u ∈ V ∀ i, j ∈ [ k ] , i 6 = j h u i , u j i = 0 (1) ∀ u ∈ V k X i =1 k u i k 2 = 1 (2) ∀ u, v , w ∈ V ∀ i, j, l ∈ [ k ] k u i − w l k 2 ≤ k u i − v j k 2 + k v j − w l k 2 (3) ∀ u, v ∈ V ∀ i, j ∈ [ k ] k u i − v j k 2 ≤ k u i k 2 + k v j k 2 (4) ∀ u, v ∈ V ∀ i, j ∈ [ k ] k u i k 2 ≤ k u i − v j k 2 + k v j k 2 (5) 2 F or ev ery verte x u and state i w e introd u ce a v ector u i . In the intended integ r al solution u i = 1, if u has state i ; and u i = 0, otherwise. All S DP constraints are satisfied in the integ r al solution; th us this is a v alid relaxation. Th e ob jectiv e function of the SDP measures what f raction of all Unique Games constraints is not satisfie d . 2 Algorithm W e define th e e arthm over distanc e b et ween t wo sets of orthogonal v ectors { u 1 , . . . , u k } and { v 1 , . . . , v k } as follo ws: ∆( { u } i , { v } i ) ≡ min σ ( i ) ∈S k k X i =1 k u i − v σ ( i ) k 2 , here S k is the sym metric group, the group of all p erm utations on the set [ k ] = { 1 , . . . , k } . Giv en an S DP solution { u i } u,i w e d efi ne the earthmov er distance b e tw een v ertices in a n atural w ay: ∆( u, v ) = ∆( { u 1 , . . . , u k } , { v 1 , . . . , v k } ) . Arora et al. [1] pro ved that if an instance of Unique Games on an expander is almost satisfiable, then the av erage earthmo ver distance b et w een t wo vertices (defined by the SDP solution) is sm all. W e will need the follo wing corollary from their results: F or every R ∈ (0 , 1) , ther e exists a p ositive c , such that for every (1 − ε ) satisfiable instanc e of Unique Games on an exp ander gr aph G , if ε/λ G < c , then the exp e cte d e arthmo ver distanc e b etwe en two r ando m vertic es is less than R i.e. E u,v ∈ V [∆( u, v )] ≤ R . In fact, Ar ora et al. [1] sh o we d that c ≥ Ω( R/ log (1 /R )), but we will not us e this b o u nd. Moreo v er, in th e rest of th e pap er, w e fi x the v alue of R < 1 / 4. W e p ick c R , so that if ε/λ G < c R , then E u,v ∈ V [∆( u, v )] ≤ R / 4 . (6) Our algorithm transforms v ectors { u i } u,i in the SDP solution to v ectors { ˜ u i } u,i using a normal- ization tec h nique in tro duced by Chlam tac, Mak aryc hev and Mak aryc hev [3]: Lemma 2.1. [3] F or every SDP solution { u i } u,i , ther e exists a set of ve ctors { ˜ u i } u,i satisfying the fol lo wing pr op erties: 1. T riangle ine qualities in ℓ 2 2 : for al l vertic es u , v , w in V and al l states i , p , q in [ k ] , k ˜ u i − ˜ v p k 2 2 + k ˜ v p − ˜ w q k 2 2 ≥ k ˜ u i − ˜ w q k 2 2 . 2. F or al l vertic es u, v in V and al l states i, j in [ k ] , h ˜ u i , ˜ v j i = h u i , v j i max( k u i k 2 , k v j k 2 ) . 3. F or al l non-zer o ve ctors u i , k ˜ u i k 2 2 = 1 . 3 4. F or al l u in V and i 6 = j in [ k ] , the ve ctors ˜ u i and ˜ u j ar e ortho gona l. 5. F or al l u and v in V and i and j in [ k ] , k ˜ v j − ˜ u i k 2 2 ≤ 2 k v j − u i k 2 max( k u i k 2 , k v j k 2 ) . The set of ve ctors { ˜ u i } u,i c an b e obtaine d in p olynomia l time. No w w e are ready to describ e the roun ding algorithm. The algorithm giv en an SDP solution, outputs an assignment of states (lab els) to the vertice s . Appro ximation Algorithm Input: an SDP s olution { u i } u,i of cost ε . Initialization 1. P ic k a rand om vertex u (unif orm ly distribu ted) in V . W e call th is v ertex the initial vertex . 2. P ic k a rand om state i ∈ [ k ] for u ; choose state i with probabilit y k u i k 2 . Note that k u 1 k 2 + · · · + k u k k 2 = 1. W e call i the initial state . 3. P ic k a rand om num b e r t uniformly distributed in the segment [0 , k u i k 2 ]. 4. P ic k a rand om r in [ R , 2 R ]. Normalization 5. O btain vecto r s { ˜ u i } u,i as in Lemma 2.1 . Propagation 6. F or ev ery v ertex v , • Find all states p ∈ [ k ] su c h that k v p k 2 ≥ t and k ˜ v p − ˜ u i k 2 ≤ r . Denote the set of p ’s b y S v : S v =  p : k v p k 2 ≥ t and k ˜ v p − ˜ u i k 2 ≤ r  . • If S v con tains exactly one element p , th en assign the state p to v . • Otherwise, assign an arb itrary (say , r andom) state to v . Denote by σ vw the partial mapp in g from [ k ] to [ k ] that m aps p to q if k ˜ v p − ˜ w q k 2 ≤ 4 R . Note that σ vw is w ell defined i.e. p cannot b e mapp e d to differen t states q a n d q ′ : if k ˜ v p − ˜ w q k 2 ≤ 4 R and k ˜ v p − ˜ w q ′ k 2 ≤ 4 R , then, b y the ℓ 2 2 triangle inequ alit y (see Lemma 2.1(1)), k ˜ w q − ˜ w q ′ k 2 ≤ 8 R , but ˜ w q and ˜ w q ′ are orthogonal un it vec tors, so k ˜ w q − ˜ w q ′ k 2 = 2 > 8 R . Clearly , σ vw defines a p artial matc hing b et ween states of v and states of w : if σ vw ( p ) = q , then σ w v ( q ) = p . 4 Lemma 2.2. If p ∈ S v and q ∈ S w with non-zer o pr ob ability, then q = σ vw ( p ) . Pr o of. If p ∈ S v and q ∈ S w then f or some v ertex u and state i , k ˜ v p − ˜ u i k 2 ≤ 2 R and k ˜ w q − ˜ u i k 2 ≤ 2 R , th us by the triangle inequalit y k ˜ v p − ˜ w q k 2 ≤ 4 R and by the defi n ition of σ vw , q = σ vw ( p ). Corollary 2.3. Supp o se, that p ∈ S v , then the set S w either e quals { σ vw ( p ) } or is empty (if σ vw ( p ) is not define d, then S w is empty). Particularly, if u and i ar e the initial ve rtex and state, then the set S w either e quals { σ uw ( i ) } or is empty. Thus, e v ery set S w c ontains at most one element. Lemma 2.4. F or every choic e of the initial vertex u , f or every v ∈ V and p ∈ [ k ] th e pr ob ability that p ∈ S v is at most k v p k 2 . Pr o of. If p ∈ S v , then i = σ vu ( p ) is the initial state of u and t ≤ k v p k 2 . The probability that b oth these ev ents h app en is Pr ( i ∈ S u ) × Pr  t ≤ k v p k 2  = k u i k 2 × min( k v p k 2 / k u i k 2 , 1) ≤ k v p k 2 (recall that t is a rand om r eal num b e r on the segmen t [0 , k u i k 2 ]). Denote the set of th ose ve r tices v for whic h S v con tains exactly one element by X . First, w e sho w that on a v erage X con tains a constan t fraction of all ve r tices (later w e will pro ve a muc h stronger b oun d on the size of X ). Lemma 2.5. If ε/λ G ≤ c R , then the exp e cte d size of X is at le ast | V | / 4 . Pr o of. C onsider an arbitrary v ertex v . Estimate the pr obabilit y that p ∈ S v giv en that u is the initial v ertex. Supp ose that ther e exists q such that k v p − u q k 2 ≤ k v p k 2 · R/ 2, then k ˜ u q − ˜ v p k 2 ≤ 2 k u q − v p k 2 max( k u q k 2 , k v p k 2 ) ≤ R. Th u s, q = σ vu ( p ) and k ˜ u q − ˜ v p k 2 ≤ r with probabilit y 1. Hence, if q is chosen as the initial s tate and k v p k 2 ≥ t , then v p ∈ S v . Th e probabilit y of this ev ent is k u q k 2 × min ( k v p k 2 / k u q k 2 , 1). Noti ce that k u q k 2 × min( k v p k 2 / k u q k 2 , 1) = min( k v p k 2 , k u q k 2 ) ≥ k v p k 2 − k u q − v p k 2 ≥ k v p k 2 2 . No w, consid er all p ’s f or which there exists q such that k v p − u q k 2 ≤ k v p k 2 · R/ 2. The probability that one of them b elo n gs to S v , and thus v ∈ X , is at least 1 2 X p :min q ( k u q − v p k 2 ) ≤k v p k 2 · R/ 2 k v p k 2 = 1 2 k X p =1 k v p k 2 − 1 2 X p :min q ( k u q − v p k 2 ) > k v p k 2 · R/ 2 k v p k 2 ≥ 1 2 − 1 2 × k X p =1 2 R min q ( k u q − v p k 2 ) ≥ 1 2 − ∆( { u } q , { v } p ) R . Since the av erage v alue of ∆( { u } q , { v } p ) ov er all p airs ( u, v ) is at most R/ 4 (see (6)), th e exp ect ed size of X (for random initial v ertex u ) is at least | V | / 4. 5 Corollary 2.6. If ε/λ G ≤ c R , then the size of X is g r e ater than | V | / 8 with pr ob ability gr e at e r than 1 / 8 . Lemma 2.7. Th e exp e cte d size of the cut b etwe en X and V \ X is at most 6 ε/R | E | . Pr o of. W e sh ow that the size of th e cut b e tw een X and V \ X is at most 6 ε/R | E | in the exp ectation for any choic e of the initial ve r tex u . Fix an edge ( v , w ) and estimate the pr obabilit y that v ∈ X and w ∈ V \ X . If v ∈ X and w ∈ V \ X , then S v con tains a unique state p , but S w is empt y (see Corollary 2.3) and, particularly , π vw ( p ) / ∈ S w . Th is h app ens in tw o cases: • There exists p su c h th at i = σ vu ( p ) is the initial state of u and k w π vw ( p ) k 2 < t ≤ k v p k 2 . The probabilit y of this ev ent is at most k X p =1 k u σ vu ( p ) k 2 ×      k v p k 2 − k w π vw ( p ) k 2 k u σ vu ( p ) k 2      ≤ k X p =1 k v p − w π vw ( p ) k 2 . • There exists p su ch that i = σ vu ( p ) is the initial state of u , t ≤ k v p k 2 and k ˜ u i − ˜ v p k 2 < r ≤ k ˜ u i − ˜ w π vw ( p ) k 2 . Th e pr obabilit y of this ev ent is at m ost k X p =1 k u σ vu ( p ) k 2 × k v p k 2 k u σ vu ( p ) k 2 ×      k ˜ u σ vu ( p ) − ˜ w π vw ( p ) k 2 − k ˜ u σ vu ( p ) − ˜ v p k 2 R      ≤ k X p =1 k v p k 2 × k ˜ v p − ˜ w π vw ( p ) k 2 R ≤ k X p =1 k v p k 2 × 2 k v p − w π vw ( p ) k 2 R · max( k v p k 2 , k w π vw ( p ) k 2 ) ≤ 2 R k X p =1 k v p − w π vw ( p ) k 2 . Note that the pr ob ab ility of the first even t is zero, if k w π vw ( p ) k 2 ≥ k v p k 2 ; and the probabilit y of th e second even t is zero, if k ˜ u σ vu ( p ) − ˜ v p k 2 ≥ k ˜ u σ vu ( p ) − ˜ w π vw ( p ) k 2 . Since the SDP v alue equals 1 2 | E | X ( v,w ) ∈ E k X p =1 k v p − w π vw ( p ) k 2 ≤ ε. The exp ected fraction of cut edges is at m ost 6 ε/R . Lemma 2.8. If ε ≤ min ( c R λ G , h G R/ 1000) , then with pr ob ability at le ast 1/16 the size of X is at le ast  1 − 100 ε h G R  | V | . 6 Pr o of. T he exp ected size of the cut δ ( X, V \ X ) b et ween X and V \ X is less than 6 ε/R | E | . Hence, since the graph G is an exp ander, one of the sets X or V \ X must b e small: E [min( | X | , | V \ X | )] ≤ 1 h G × E [ | δ ( X , V \ X ) | ] | E | × | V | ≤ 6 ε h G R | V | . By Mark o v’s Inequ alit y , Pr  min( | X | , | V \ X | ) ≤ 100 ε h G R | V |  ≥ 1 − 1 16 . Observe , that 100 ε/ ( h G R ) | V | < | V | / 8. Ho w ev er, by Corollary 2.6, the size of X is g r eater than | V | / 8 with probabilit y greater than 1 / 8. T h u s Pr  | V \ X | ≤ 100 ε h G R | V |  ≥ 1 16 . Lemma 2.9. The pr ob ability that for an arbitr ary e dge ( v , w ) , the c onstr aint b etwe e n v and w is not satisfie d, but v and w ar e i n X is at most 4 ε vw , wher e ε vw = 1 2 k X i =1 k v i − w π vw ( i ) k 2 . Pr o of. W e show that for ev ery c h oice of the initial vertex u the desir ed probability is at m ost 4 ε vw . Recall, that if p ∈ S v and q ∈ S w , then q = σ vw ( p ). The constr aint b e tw een v and w is n ot satisfied if q 6 = π vw ( p ). Hence, the pr obabilit y that the constrain t is not satisfied is at most, X p : π vw ( p ) 6 = σ vw ( p ) Pr ( p ∈ S v ) . If π vw ( p ) 6 = σ vw ( p ), then k ˜ v p − ˜ w π vw ( p ) k 2 ≥ k ˜ w π vw ( p ) − ˜ w σ vw ( p ) k 2 − k ˜ v p − ˜ w σ vw ( p ) k 2 ≥ 2 − 4 R ≥ 1 . Hence, by L emm a 2.1 (5), k v p − w π vw ( p ) k 2 ≥ k v p k 2 / 2 . Therefore, by Lemma 2.4, X p : π vw ( p ) 6 = σ vw ( p ) Pr ( p ∈ S v ) ≤ X p : π vw ( p ) 6 = σ vw ( p ) k v p k 2 ≤ 2 k X p =1 k v p − w π vw ( p ) k 2 = 4 ε vw . Theorem 2.10. Ther e exists a p olynomial time ap pr oximation algorithm tha t giv en a (1 − ε ) satisfiable instanc e of Unique Games on a d -exp ander gr aph G with ε/λ G ≤ c , the algorithm finds a solution of c ost 1 − C ε h G , wher e c and C ar e some p ositive absolute c onstants. 7 Pr o of. W e describ e a randomized p olynomial time algorithm. Ou r algorithm ma y return a solution to the SDP or output a sp ec ial v alue fail . W e sho w that the algorithm outputs a s olution with a constan t pr ob ab ility (that is, the probabilit y of f ailure is b o u n ded a wa y from 1); and conditional on the even t that the algorithm outpu ts a solution its exp e cted v alue is 1 − C ε h G . (7) Then w e argue that the algorithm can b e easily derand omized — simply b y en umerating all p ossible v alues of the random v ariables used in the algorithm and pic king the best solution. Hence, the deterministic algorithm fin d s a solution of cost at least (7). The randomized algo r ithm first solv es the SDP an d then ru ns the rounding pro cedure describ ed ab o ve. If the size of the set X is more th an  1 − 100 ε h G R  | V | , the algorithm outputs the obtained solution; otherwise, it outputs fail . Let us analyze the algorithm. By Lemma 2.8, it su cceeds with probabilit y at least 1 / 16. T he fraction of edges having at least on e end p oint in V \ X is at most 100 ε/ ( h G R ) (since the graph is d - regular). W e conserv ativ ely assu me that the constraints corresp onding to these edges are violated. The exp ected num b er of violated constraints b et ween vertice s in X , by Lemma 2.9 is at most 4 P ( u,v ) ∈ E ε uv Pr ( | X | ≥ 100 ε / ( h G R )) ≤ 64 ×   1 2 X ( u,v ) ∈ E k u i − v π vw ( i ) k 2   ≤ 64 ε | E | . The total f raction of violated constrain ts is at m ost 100 ε/ ( h G R ) + 64 ε . References [1] S. Arora, S. Khot, A. Kolla, D. Steurer, M. T ulsiani, and N. Vishnoi. Ne ar-Optimal Algorithms for U nique Games. In P r o ceedings of the 40th A CM Sym p osium on T h eory of C omp uting, pp. 21–28, 2008. [2] M. Charik ar, K . Mak aryc hev, and Y. Mak aryc hev. Ne ar-Optimal Algorithms for U nique Games. In Pr o ceedings of the 38th ACM Symp osium on Theory of Compu ting, p p. 205–214, 2006. [3] E. Chlam tac, K. Mak aryc hev, and Y. Mak aryc hev. Unique games on exp anding c onstr aint gr aph s ar e e asy. In Pro c eedings of the 47th IEEE Symp osium on F ound ations of Compu ter Science, p p. 687–6 96, 2006 . [4] A. Gupta and K. T a lwar. Appr oxima ting Unique Games. In Pro ceedings of the 17th ACM- SIAM S ymp osium on Discrete Algorithms, pp. 99–106, 2006. [5] S. Khot. On the p o wer of unique 2-pr over 1-r ound games . In Pro c eedings of the 34th ACM Symp o siu m on Theory of C omputing, pp. 767–77 5, 2002. 8 [6] S. Khot, G. Kindler, E. Mossel, and R. O’Donnell. Optimal inappr oximability r esults for MAX-CU T and other two-variable CSPs? SIAM Journ al of Comp uting 37(1), pp. 319–35 7, 2007. [7] L. T r evisan. Appr oxima tion Algorithms for Unique Games. In Pro ceedings of the 46th IEEE Symp o siu m on F oundations of Computer Science, pp . 197–205, 2005. 9