Max Cut and the Smallest Eigenvalue

Max Cut and the Smallest Eigen v alue Luca Trevisan ∗ No v em b er 26, 2024 Abstract W e describ e a new a pproximation algorithm for Max Cut. Our algorithm runs in ˜ O ( n 2 ) time, where n is the n umber of v ertices, and achiev es an approximation ratio of . 531 . On insta nce s in whic h an optimal solution cuts a 1 − ε fraction of edges, our a lgorithm finds a solution that cuts a 1 − 4 √ ε + 8 ε − o (1) fraction of edges . Our main result is a v a riant of spectral par titioning, whic h can be implemented in n ear ly linear time. Given a graph in which the Max Cut o ptim um is a 1 − ε fraction of edg es, our sp e c tr al par titioning algor ithm finds a set S of vertices and a bipa rtition L, R = S − L of S such that at least a 1 − O ( √ ε ) fra ction of the edges incident o n S hav e one endp o in t in L and o ne endpo in t in R . (This can b e seen as a n ana log of Cheeger ’s inequality for the smallest eigenv alue of the adjacency matrix of a gra ph.) Iterating this pro cedure yields the approximation results stated ab ov e. A different, more complicated, v ariant of sp ectral partitioning leads to an ˜ O ( n 3 ) time a lgo- rithm that cuts 1 / 2 + e − Ω(1 /ε ) fraction of edges in g raphs in which the optim um is 1 / 2 + ε . 1 In tro duction In the Max CUT prob lem, we are giv en an undirected graph with non-n egativ e weig hts on the edges and we wish to find a partition of the v ertices (a cu t ) whic h maximizes the w eigh t of edges whose endp oints are on different sides of the partition (suc h edges are said to b e cut b y the partition). W e refer to the c ost of a solution as the fraction of w eigh ted edges of the graph that are cu t by the solution. It is easy , give n any graph, to fi nd a solution that cuts half of the edges, provi ding an appr o ximation factor of 1 / 2 for the p roblem. The algorithm of Go emans and Williamson [ GW95 ], based on a Semidefinite Programming (SDP) r elaxation, has a p erformance r atio of . 878 · · · on general graphs, and it fin ds a cut of cost 1 − O ( √ ε ) in graphs in whic h the optim um is 1 − ε . Assuming the unique games conjecture, b oth r esults are b est p ossible for p olynomial time algorithms [ Kho02 , KK MO04 , MOO05 ] (see also [ O W08 ]). Arora and Kale [ AK07 ] sho w that the Goemans-Williamson SDP relaxatio n can b e near-optimally solv ed in n early linear time in graphs of b ounded degree (or more generally , in w eigh ted graph s with b ounded ratio b et w een largest and smallest degree). W e sh o w in App endix A.1 that, using a r eduction [ T r e01 ], the Arora-Kale algorithm can b e used to ac hiev e the ∗ luca@cs.be rkeley.edu . U.C. Berkel ey , Computer Science Div ision. This material is based up on work supp orted by the National Science F ound ation under grant N o. CCF-0729 137 and by the BSF under grant 2002246. 1 appro ximation p erformance of the Go emans-Williamson algorithm on all graphs in nearly-linear time. A different r ou n ding algorithm for the Goemans-Williamson relaxation, d ue to C harik ar and Wirth [ CW04 ], finds a solution that cuts at least a 1 / 2 + Ω( ε/ log 1 /ε ) f raction of edges in graphs in whic h the optimum is 1 / 2 + ε . T his r esult to o is tight, assuming the unique games conjecture [ K O06 ]. No metho d other than SDP is kno wn to yield an appro ximation b etter than 1 / 2 for Max C ut, and su c h appr o ximation has b een ru led out for large classes of Linear Programming Relaxatio ns [ dlVKM07 , ST T 07 ]. A main source of difficult y in designing appro ximation algorithms for max cut is the lac k of go o d upp er b ound tec hniques for the max cut optim um of general graph . Indeed, sup p ose that one is able to design and analyse a n ew p olynomial-time alg orithm for max cut ac hieving, sa y , a . 51 appro ximation ratio, and consider the b eha viour of the algorithm when giv en a graph wh ose max cut optim um is . 501. Then the algorithm will clearly output a cut of cost ≤ . 501, but then the computations p erformed by the algorithm, plus the p ro of of its appro ximation r atio, provide a c ertific ate that the optimum cut in the giv en graph is ≤ . 50 1 /. 51 < . 983. The problem is that, except for semidefin ite p rogramming, we know of no tec hnique th at can pr ovide, for eve ry graph of max cut optim um ≤ . 501, a certificate that its optim u m is ≤ . 99. Indeed, the r esu lts of [ dlVKM07 , STT 07 ] show that large classes of Linear Programming relaxatio ns of max cut are unable to d istinguish suc h instances. It is p ossible, ho we ve r, to deve lop a new appro ximation algorithm that u s es semidefin ite p r ogram- ming only in the analysis, by s h o wing that if the algorithm outputs a cut of cost c , then there is a dual solution for the Go emans-Williamson SDP relaxatio n of cost at most c/. 51 , th us proving that the max cut optim um is at most c/. 51 and that the algorithm h as a p erf orm ance ratio at least . 51. Suc h primal-dual algorithms, wh ic h u se a relaxation only in the analysis, ha v e b een deriv ed for sev eral problems based on Line ar Pr o gr amming relaxations, bu t unfortunately , as discussed ab ov e, linear programming relaxations are un lik ely to b e helpfu l in max cut ap p ro ximation. As far as w e kno w, the only examples of primal-dual approximat ion algorithms f or com binatorial p roblems based on S emidefinite Pr ogramming are the algorithms for the sparsest cut problem describ ed in [ AR V04 , KR V06 , OSVV08 ]. Our R esults Our main result is a v arian t of the sp ectral partitioning algorithm w ith the follo wing p rop ert y: giv en a graph G = ( V , E ) in which the Max CUT optim um cost is 1 − ε , it find s a set S and a partition of S int o t wo disjoint sets of v ertices L, R suc h that the n umb er of edges w ith one end p oin t in L and one endp oin t in R is at least a 1 − O ( √ ε ) of the total num b er of edges in cident 1 on S . More pr ecisely , we show that the n umb er of edges ha ving b oth endp oints in L or b oth endp oin ts in R , plus half the num b er of edges ha ving an end p oin t in S and an end p oin t in V − S is at most a 2 √ ε + o (1) fraction of the edges incident on S . (See Theorem 1 and the su bsequen t discussion.) W e will ignore the o (1) additiv e factors in the su bsequen t discussion in this section. T o d eriv e an appro ximation algorithm for Max CUT, giv en a graph we apply the partitioning algorithm and fi nd sets L, R as ab o v e, we remo v e the vertic es in L ∪ R from the graph, recursiv ely 1 An edges ( i, j ) is incident on a set S of vertices if at least one of the en dp oin ts i, j b elongs to S . 2 find a partition of the residual graph, and then put bac k the ve rtices of L on one side of the partition and v ertices of R on the other side. This means that we cut all the ed ges that are cut in the recursiv e step, plus all the edges with one endp oint in L and one endp oint in R , plus at least half of the edges b et wee n S and V − S . The recur s ion is stopp ed when less than half of the edges inciden t on S are cut, in wh ic h in case we return a greedy partition of the residual graph. W e present an analysis of the recursiv e pro cedure due to Moses Charik ar, whic h improv es an analysis of ours wh ic h app eared in a previous version of this pap er. The f ollo wing observ ation pla ys an imp ortant role in th e analysis: at a generic step of the execution of the algorithm, if the optimal solution in the original graph is 1 − ε , and the current residual graph h olds a ρ fr actio n of the original edges, then we kno w that the optim um in the curren t r esidual graph is at least 1 − ε/ ρ , and th e sp ectral algorithm cuts at least a 1 − 2 p ε/ρ fraction of the edges inciden t on L ∪ R . When the recursion ends, it is b ecause the sp ectral algorithm cuts less than half of the edges inciden t on L ∪ R , and so the optim um of the r esidual graph at the end of the recursion must b e less than 15 / 16 , meaning that the r esidu al graph at the end of the recursion conta ins at most a 16 ε fraction of the edges of the original graph. Putting together this information, a calculation shows that the algorithm cuts at least a 1 − 4 √ ε + 8 ε fr actio n of edges of the graph. The ratio (1 − 4 √ ε + 8 ε ) / (1 − ε ) is alw a ys at least . 531. When applied to graphs in which the optimum is close to 1 / 2 (in f act, to any graph in which the optim um is sm aller th an 15 / 16), our algorithm ma y simply return a greedy partition. T hus, it fails to pr o vide any non-trivial approxima tion to the Max C u tGain p roblem, which is the same as the Max Cut problem, except that we coun t the num b er of cut edges minus | E | / 2. (Equiv alen tly , w e coun t the n umb er of cut edges minus the num b er of uncut edges.) F or Max CutGain we dev elop a more sophisticated sp ectral partitioning algorithm with the follo wing p rop ert y: giv en a graph in whic h the Max Cut optimum is 1 / 2 + ε , our algorithm finds sets L, R suc h that the num b er of edges incident on L ∪ R cut by the partition exceeds the num b er of u n cut ed ges b y at least a 1 /exp (Ω(1 / ε )) fraction of the edges incident on L ∪ R . Iterating this algorithm allo ws us to find a cut for th e en tire graph of cost at least 1 / 2 + 1 /exp (Ω(1 /ε )). This second algorithm can b e also app lied to the case in whic h edges ha ve negativ e we igh ts, and it appro ximates a general class of quadr atic pr ograms. Give n a symmetric r eal-v alued matrix Q with zero es on the diagonal, if there exists a v ector x ∈ {− 1 , 1 } V suc h that x T Qx ≥ ε · || Q || 1 , our algorithm fi nds a v ector y ∈ {− 1 , 1 } V suc h that y T Qy ≥ exp ( − O (1 /ε )) · || Q || 1 , where || Q || 1 := P i,j | Q ( i, j ) | . (The algorithm of Ch arik ar and Wirth find s a vec tor y su ch that y T Qy ≥ || Q || 1 · ε/ log 1 /ε .) Relation to Cheeger’s Inequalit y In the case of regular graphs, our main result, Th eorem 1 , ma y b e s een as an analog of Ch eeger’s inequalit y [ Alo86 ] for the smallest (rather than second largest) eigen v alue of the adjacency m atrix of the graph. W e discuss this analogy in S ectio n 5 Relation to the Go emans-Williamson Relaxation Our algorithm m a y also b e seen as a primal-dual algorithm tha t prod uces, along with a cut, a feasible solution to the semidefi n ite dual of the Go emans-Willi amson relaxation suc h that the cost 3 of the cut is at least . 531 times the cost of the d ual solution. W e d escrib e this view in S ectio n 6 . Other Relations to Previous W ork It has b een known that one can u s e s p ectral metho ds to certify an upp er b ound to the Max CUT optim um of a giv en graph. I n particular, if G is a d -regular graph of adjacency matrix A , and M := 1 d A has eigen v alues 1 = λ 1 ≥ λ 2 ≥ · · · ≥ λ n , then on e can easily sh ow 2 that Max Cut ≤ 1 2 + 1 2 | λ n | (1) (Our Lemma 2 is essenti ally a restatemen t of this fact.) What is new is that w e are able to prov e a c onverse , in Lemma 3 , and sho w that a non-trivial consequence follo ws wheneve r | λ n | is close to 1. As mentioned ab ov e, it was known that λ n = − 1 if and only if G has a bipartite connected comp onen t. In particular, if G is connected and not bipartite then λ n > − 1. Alon and Su dak o v [ AS00 ] consider the qu estion of h o w small, in su c h case, can the gap 1 − | λ n | b e. T h ey sho w that, if G is co nnected and not bipartite, it has maxim um degree d and diamet er D , and λ n is the smallest eigen v alue of th e adj acency matrix A , then d − | λ n | ≥ 1 ( D +1) · n . Th e b oun d was imp r o v ed to d − | λ n | ≥ 1 D · n b y Cioaba [ Cio07 ]. Our result implies the wea ke r b ound d − | λ n | ≥ 1 dn 2 in a d -regular graph. The “con v erse expander mixing lemma” of Bilu and Linial [ BL06 ] has some similarit y with our approac h to Max CutGain. Bilu and Linial show that if G is a d -regular graph, A is the adjacency matrix, and λ 1 ≥ · · · ≥ λ n are the eigen v alues of M := 1 d A , then if m ax { λ 2 , | λ n |} ≥ ε it follo ws that there are s ets L, R suc h that the num b er of edges b et w een L an d R differs from wh at one w ould exp ect in a rand om d -regular graph b y a multi plicativ e error factor Ω( ε/ log 1 /ε ). In our main resu lt for Max CutGain (Theorem 11 ) w e ha ve a stronger assumption, that | λ n | ≥ ε , but we need to derive a muc h s tr onger conclusion, namely that the num b er of edges b et w een L and R not only exceeds the num b er of edges that one would exp ect in a r and om d -regular graph (a fact that can b e probably pro ve d with the s ame quantita tiv e result of Bilu-Linial), bu t in fact exceeds the n umber of edges wh ic h are en tirely con tained in L or entirely con tained in R . The main difference b et w een our pro of and the pr o of of Bilu and Linial is that the com bina- torial quant it y that they relate to max { λ 2 , | λ n |} is the optim um of the normalized multil inear form max x,y ∈{− 1 , 0 , 1 } | x T M y | / ( || x || · || y || ), for a certain matrix M , wh ile the com binatorial qu an- tit y that we w ish to relate to | λ n | is the optimum of the normalized homogeneous quadratic form max x ∈{− 1 , 0 , 1 } | x T M x | / || x || 2 , for a differen t matrix M . Generally , it is considerably harder to round con tin uous relaxations of qu adratic forms of the latter t yp e compared to m ultilinear forms of the first kind. (See e.g. the introduction of [ CW04 ] and their discussion of their results v ersus the results of Alon and Naor [ AN06 ].) 2 Inequality ( 1 ) appears to b e a folklore result. Lov´ asz [ Lov03 , Prop osition 4.4] credits it t o Delorme and P oljac k [ DP93a , DP93b ]. The earliest related reference we are aw are of is [ Hae79 , Theorem 2.1.4.i], which states that if V 1 , V 2 is a partition of a d -regular graph G = ( V , E ), and if d 1 is the av erage degree of the subgraph induced by V 1 , then n 1 d − nd 1 ≤ − λ n · ( n − n 1 ), from which one can d eriv e that n 1 · ( d − d 1 ), the num b er of ed ges crossing th e cut, ob eys n 1 · ( d − d 1 ) ≤ n 1 · ( n − n 1 ) · ( d − λ n ) /n , and the latter term is at most n · ( d − λ n ) / 4. 4 The id ea of iterativ ely remo ving p arts of an instance in w h ic h one has a go o d solution app ears in v arious works on the sp ars est cut problem (for example in the wa y Spielman and T en g [ ST04 ] find a balanced separator us in g their “nibble” pro cedure), and it w as u sed to approximat e the Max CUT problem (in the ve rsion in wh ic h one wa nts to minimize the n umber of u ncut ve rtices) by Agarw al et al. [ A CMM05 ]. In the algorithm of Agarw al et al., as in our algorithm, the basic pro cedur e that is b eing iterated fi n ds a set S of vertice s and a bip artition L, R of S such th at most of the edges inciden t on S ha v e one endp oint in L and one end p oin t in R . 2 Sparsification It follo ws from the C hernoff Boun d that if w e are giv en a graph G = ( V , E ) and w e sample O ( δ − 2 | V | ) edges with replacemen t 3 then, w ith high pr obabilit y , eve ry cut ( S, ¯ S ) h as the same cost in the original graph as in the new graph, up to an add itive error δ . 4 F or this r eason, all the d ep endency on | E | in the run ning time of our algorithm can b e changed to a dep endency on | V | with an arbitrarily small loss in the appro ximation factor. 3 The Sp ectral Algorit hm In this section w e p r o v e our main resu lt. Theorem 1 (Main) Ther e is an algorithm that, given a gr aph G = ( V , E ) for which the optimum of the Max CUT pr oblem is at le ast 1 − ε , and a p ar ameter δ , finds a ve ctor y ∈ {− 1 , 0 , 1 } V such that P i,j A i,j | y i + y j | P i d i | y i | ≤ 4 √ ε + δ wher e A i,j is the weight of e dge ( i, j ) and d i is the (weighte d) de gr e e of v ertex i . The algorithm c an b e implemente d in ne arly-line ar r andomize d time O ( δ − 2 · ( | V | + | E | ) · log | V | ) . T o und ers tand the statemen t of Theorem 1 , let y b e the v ector r eturned by the algorithm, and call L the set of v ertices w ith negativ e co ordinates in y , and R the set of v ertices with p ositiv e co ord inates. Then, up to constan t factors, the numerato r counts the n umber of edges incident on L ∪ R wh ic h f ail to h a v e one endp oint in L and one end p oin t in R , the denominator count s th e n umber of incident incident on S . Mo re sp ecifically , the n umerator count s four times the edges that are en tirely con tained in L or entirel y cont ained in R , and t wice the edges that ha ve one endp oint in S an d one endp oin t in V − S . The denominator count s ev ery edge inciden t on L ∪ R once or t wice, dep ending on whether one or b oth the endp oin ts of the edge are in S . 3 If th e graph is unw eighte d, we sample from the un iform distribution o ver the edges; otherwise we sample from the distribution in which each edge has a probability prop ortional to its w eight. 4 Note th at the sparsified graph is an u nwe ighted m ultigraph, and that the sp arsification pro cess is considerably simpler than the one u sed for algorithms for sparsest cut and other graph minimization problems. 5 The follo wing form of the conclusion of Theorem 1 will b e con v enien t in our analysis: giv en the v ector y , call M the num b er of edges inciden t on L ∪ R , U the num b er of “un cu t” edges that ha v e b oth endp oin ts in L or b oth endp oints in R , and X the n umb er of “cross” edges that h a v e exactly one endp oin t in L ∪ R ; then U + 1 2 X ≤  2 √ ε + δ 2  · M Let A b e the adjacency matrix of our input graph G (hence A i,j is the weigh t of the edge b et we en i and j ), and D b e the diagonal matrix such that D i,i is the we igh ted degree d i of vertex i and D i,j = 0 for i 6 = j . Theorem 1 follo ws by com bining the f ollo wing tw o r esults, and noting that, for a, b ≥ 0, √ a + b ≤ √ a + √ b . Lemma 2 If the optimum Max CUT in G has c ost at le ast 1 − ε , ther e is a ve ctor x ∈ R V such that x T ( D + A ) x ≤ 2 ε · x T D x . F u rthermo r e, for eve ry δ > 0 , we c an find in time O ( δ − 1 · ( | E | + | V | ) · log | V | ) a ve ctor x ∈ R V such that x T ( D + A ) x ≤ (2 ε + δ ) · x T D x Lemma 3 Given a ve ctor x ∈ R V such that x T ( D + A ) x ≤ ε · x T D x , we c an find i n time O ( | E | + | V | log | V | ) a ve c tor y ∈ {− 1 , 0 , 1 } V such that P i,j A i,j | y i + y j | P i d i | y i | ≤ √ 8 ε (2) Lemma 2 has a simp le pro of, and it can be seen as a statemen t about th e semidefinite du al of the Go emans-Willia mson r elaxat ion, as discussed in Section 6 . Lemma 3 is the main result of this pap er. 3.1 Pro of of Lemma 2 Consider the optimization problem min x ∈ R V x T Ax x T D x (3) Let ( S, ¯ S ) b e an optimum cut for G , and defin e the vec tor x ∗ ∈ {− 1 , 1 } V suc h that x ∗ i = 1 if i ∈ S and x ∗ i = − 1 otherwise. Then x ∗ T Ax ∗ equals twic e the d ifference b et wee n the num b er of edges not cut by ( S, ¯ S ) and the num b er of edges that are cut, wh ich is at most 2 · (2 ε − 1) · | E | . As for x ∗ T D x ∗ , we h a v e x ∗ T D x ∗ = X i d i · ( x ∗ i ) 2 = X i d i = 2 · | E | 6 Th us x ∗ is a feasible s olution to ( 3 ) of cost at most 2 ε − 1, and if ˆ x is the op timal solution to ( 3 ), then we must h a v e ˆ x T A ˆ x ≤ (2 ε − 1) ˆ x T D ˆ x T o p ro v e the “furthermore” part of the lemma, w e observe that the optimization problem in ( 3 ) is equiv alen t to min x ∈ R V x T D − 1 / 2 AD − 1 / 2 x x T x (4) where D − 1 / 2 is the matrix that such that D − 1 / 2 i,j = 0 if D i,j = 0, and D − 1 / 2 i,j = 1 / p D i,j otherwise. In turn, the optimization pr oblem in ( 4 ) is the p roblem of computing the smallest eigen v alue of D − 1 / 2 AD − 1 / 2 , whic h is the same as compu ting the largest eigen v alue of the p ositiv e semidefinite matrix I − D − 1 / 2 AD − 1 / 2 . Giv en a n × n p ositiv e semidefinite matrix M with T non-zero en tries and of largest eigen v alue λ 1 , and a parameter δ , it is p ossible to fi n d a vect or x such that x T M x ≥ λ 1 · (1 − δ ) · x T x in randomized time O ( δ − 1 · ( T + n ) · log n ) [ KW92 ]. Applying the algorit hm to I − D − 1 / 2 AD − 1 / 2 , whic h, as prov ed ab ov e, has a largest eigen v alue wh ic h is at least 2 − 2 ε , and wh ic h has | E | + | V | non-zero ent ries, w e fi n d in randomized time O ( δ − 1 / 2 · ( | E | + | V | ) · log | V | ) a v ector x ′ suc h that x ′ T ( I − D − 1 / 2 AD − 1 / 2 ) x ′ x ′ T x ′ ≥ 2 − 2 ε − δ and so x ′ T D − 1 / 2 AD − 1 / 2 x ′ ≤ (2 ε + δ − 1) · x ′ T x ′ and, if we define x ′′ := x ′ D 1 / 2 , then x ′′ T Ax ′′ ≤ (2 ε + δ − 1) x ′′ T D x ′′ whic h we can rewrite x ′′ T ( A + D ) x ′′ ≤ (2 ε + δ ) x ′′ T D x ′′ 3.2 Pro of of Lemma 3 W e no w come to our main resu lt. The condition x ( D + A ) x ≤ ε · xD x is equiv alen t to 1 2 X i,j A i,j ( x i + x j ) 2 ≤ ε X i d i x 2 i (5) Before starting the f orm al pr o of, we d escrib e a heuristic argumen t that giv es some int uition for the actual pro of. 7 Pr o of Ide a. Equation ( 5 ) states that the a v erage v alue of ( x i + x j ) 2 , for an edge ( i, j ), is at most ε times the a v erage v alue of x 2 i and x 2 j . So, non-rigorously , we would guess that for a typica l edges the v alue of | x i + x j | is at most ab out √ ε times | x i | + | x j | . F or this to h app en, it m ust b e the case th at x i and x j ha v e differen t s igns, and their absolute v alue is nearly the same; that is, for some p ositiv e c , x i = − c and x j = c (1 − √ ε ). Supp ose no w that w e pic k a random threshold t , and we defin e y i = − 1 ⇔ x i ≤ − t and y i = 1 ⇔ x i ≥ t . T h en | y i − y j | is 2 with probabilit y c √ ε and zero otherwise, while | y i | and | y j | are 1 with probabilit y roughly c and zero otherwise; then it follo ws th at the exp ectation of P ( i,j ) | y i + y j | is ab out a √ ε fraction of the exp ectati on of P i d i | y i | . Our algorithm, which we call the 2-Thresholds Sp ectral Cut algorithm and abb r eviate 2TSC, is as follo ws: • Algorithm 2TSC • F or ev ery v ertex k – Define the v ector y k ∈ {− 1 , 0 , 1 } V as follo ws: y k i = − 1 iff x i < − | x k | y k i = 1 iff x i > | x k | y k i = 0 iff | x i | ≤ | x k | • Output the vect or y k for whic h the ratio P i,j A i,j | y k i + y k j | P i d i | y k i | is smallest The algorithm can b e implemente d to run in O ( | E | + | V | log | V | ) time. W e fi rst sort the ve rtices according to the v alue of | x i | , and so w e assume we ha v e | x 1 | ≤ | x 2 | ≤ · · · ≤ | x n | when w e ru n 2TSC. At eac h step k , we need to mo dify the v ector y only in p ositions k and k − 1, and the cost of recomputing the ration is only O ( d k + d k − 1 ), so that all the n steps together take time O ( | E | ). W e need to argue that, under the assump tion of the L emma, the algorithm outputs a v ector y suc h that the ratio in ( 2 ) is at most √ 8 ε In order to analyze 2TSC, we study the f ollo win g r andomized pro cess: • Pic k a v alue t uniform ly in [0 , max i x 2 i ]; • Define Y ∈ {− 1 , 0 , 1 } V as follo ws: Y i = − 1 iff x i < − √ t Y i = 1 iff x i > √ t Y i = 0 iff | x i | ≤ √ t 8 Ev ery Y that is generated b y the pr ob abilistic pro cess with p ositiv e probabilit y is considered b y algorithm 2 TSC at some stage; this implies that if algo rithm 2TSC outputs a ve ctor y suc h that P i,j A i,j | y ( i ) + y ( j ) | > √ 8 ε P i d i | y i | , then in the randomized pro cess w e m ust ha ve P i,j A i,j | Y ( i ) + Y ( j ) | > √ 8 ε P i d i | Y i | with probabilit y 1 and , in particular, E P i,j A i,j | Y ( i ) + Y ( j ) | > √ 8 ε E P i d i | Y i | . W e shall pr o v e that E P i,j A i,j | Y ( i ) + Y ( j ) | ≤ √ 8 ε E P i d i | Y i | and so w e sh all conclude that the output of algorithm 2TSC satisfies the Claim. Since Equation ( 5 ) and the distribution Y are inv arian t under m ultiplying x by a scalar, we ma y assume that max i | x i | = 1, so that t is chose n uniform ly in [0 , 1]. A case analysis sho ws th at, for ev ery edge ( i, j ), E | Y i + Y j | ≤ | x i + x j | · ( | x i | + | x j | ) (6) T o v erify Equ ation ( 6 ) we need to d istinguish the case in which x i and x j ha v e d ifferen t signs from the case in whic h they hav e the same sign. W e assu me without loss of generalit y th at | x i | > | x j | . • If they hav e different signs, and , say , | x i | > | x j | , then | Y i + Y j | = 1 when | x j | 2 ≤ t ≤ | x i | 2 , and zero otherwise. Ind eed, if t < | x j | 2 , then Y i = − Y j and | Y i + Y j | = 0, and if t > | x i | 2 then Y i = Y j = 0. So E | Y i + Y j | equals | x i | 2 − | x j | 2 , whic h is equal to the right -hand side of Equation ( 6 ). • If they hav e the same sign, then | Y i + Y j | = 2 when t ≤ | x j | 2 , | Y i + Y j | = 1 wh en | x j | 2 < t ≤ | x i | 2 , and | Y i + Y j | = 0 when t > | x i | 2 . Ov erall, E | Y i + Y j | equals 2 x 2 j + ( x 2 i − x 2 j ) = x 2 j + x 2 i . The righ t-hand-sise of Equation ( 6 ) is ( x i + x j ) 2 , wh ich is only larger. Note also that E | Y i | = x 2 i . T o complete our argumen t it remains to apply Cauch y-S c h w arz and standard m anipulations. E X i,j A i,j | Y i + Y j | ≤ X i,j A i,j | x i + x j | · ( | x i | + | x j | ) ≤ s X i,j A i,j | x i + x j | 2 · s X i,j A i,j ( | x i | + | x j | ) 2 By our assumption, X i,j A i,j | x i + x j | 2 ≤ 2 ε X i d i x 2 i and it is a standard calculation that X i,j A i,j ( | x i | + | x j | ) 2 ≤ 2 X i,j A i,j ( | x i | 2 + | x j | 2 ) = 4 X i d i x 2 i 9 and so E X i,j A i,j | Y i + Y j | ≤ √ 8 ε X i d i x 2 i = √ 8 ε E X i d i | Y i | This completes the pro of that Algorithm 2TSC p erforms as required by the Lemma. 4 Appro ximation for Max Cut In this section w e analyze the f ollo wing algorithm • Algorithm: R ecursive-Spec tral-Cut • Input: graph G = ( V , E ), accuracy parameter δ • Run th e algorithm of Theorem 1 with accuracy p arameter δ , and let y ∈ {− 1 , 0 , 1 } b e the solution foun d by the algorithm; call M the w eigh ted num b er of edges ( i, j ) such that least one of y i or y j is non-zero, C the w eigh ted num b er of cut edges ( i, j ) su ch that y i , y j are b oth non-zero and hav e opp osite signs, and X the weigh ted n umb er of cr oss edges ( i, j ) such that exactly one of y i , y j is zero; • If C + 1 2 X ≤ 1 2 M , then fi nd a partition of V that cuts ≥ | E | / 2 edges, and return it. • If C + 1 2 X > 1 2 M , th en let L := { i : y i = − 1 } , R := { i : y i = 1 } , V ′ := { i : y i = 0 } , let G ′ = ( V ′ , E ′ ) b e the graph induced by V ′ , recursivel y call Recu rsive-Spectra l-Cut on G ′ , and let V 1 , V 2 b e the partion foun d by th e algorithm; return ( V 1 ∪ L, V 2 ∪ R ) or ( V 1 ∪ R, V 2 ∪ L ), whic hev er is b etter. Note that the algorithm r uns in randomized time O ( δ − 2 · | V | · ( | V | + | E | ) · log | V | ) b ecause eac h iteratio n take s time O ( δ − 1 · ( | V | + | E | ) · log | V | ) and there are at most | V | iterations. In a pr eliminary v ersion of this pap er we pr esen ted a simple argument s ho wing that if opt ≥ 1 − ε , then the algorithm cuts at least 1 − O ( ε 1 / 3 ) − δ fraction of edges. The follo wing tigh ter argument is due to Moses Charik ar (p ersonal comm unication, July 2008). Theorem 4 If Algorithm Recursive -Spectral-Cut r e c eives in input a gr aph G = ( V , E ) whose optimum is 1 − ε , with ε < 1 / 16 then it finds a solution that cuts at le ast a 1 − 4 √ ε + 8 ε − δ 2 fr action of e dges. Pr oof: Consider the t -th ite ration of the alg orithm, and let G t b e the residual graph at that iteratio n, and let ρ t · | E | b e the num b er of edges of G t . Then we observ e that the Max Cut optim um in G t is at least 1 − ε/ρ t . Let S t b e the set of v ertices and L t , R t the p artition foun d by the algorithm of T heorem 1 . Let G t +1 b e the residu al graph at the f ollo wing step, and ρ t +1 · | E | the num b er of edges of G t +1 . (If the algorithm stops at the t -th iteration, we shall take G t +1 to b e the empty graph; if the algorithm 10 discards L t , R t and c ho oses a greedy cu t, we shall tak e G t +1 to b e empty and L t , R t to b e the partition giv en by the greedy cut.) W e kno w by Theorem 1 that the algorithm will cut at least a 1 − 2 p ε/ρ t − δ / 2 fraction of the | E | · ( ρ t − ρ t +1 ) edges inciden t on S t . Indeed, we kno w that at least a max { 1 / 2 , 1 − 2 p ε/ρ t − δ / 2 } fr actio n of those edges are cut (for small v alue of ρ t , it is p ossible that 1 − 2 p ε/ρ t + δ / 2 < 1 / 2, bu t the algorithm is alw a ys guaran teed to cut at least h alf of the edges incident on S t ). Th is means that any con v ex com bination of 1 / 2 and 1 − 2 p ε/ρ t − δ / 2 is still a low er b ound on the fraction of edges incident on S t cut b y the algorithm. If b oth ρ t and ρ t +1 are at least 16 ε , we are going to use the lo w er b ound | E | · ( ρ t − ρ t +1 ) ·  1 − 2 r ε ρ t − δ 2  = | E | Z ρ t ρ t +1  1 − 2 r ε ρ t − δ 2  dr ≥ | E | Z ρ t ρ t +1  1 − 2 r ε r + δ 2  dr If ρ t ≥ 16 ε ≥ ρ t +1 , then we use the lo wer b oun d | E | · ( ρ t − 16 ε ) ·  1 − 2 r ε ρ t + δ 2  + | E | · (16 ε − ρ t +1 ) · 1 2 ≥ | E | Z ρ t 16 ε  1 − 2 r ε r − δ 2  dr + | E | · Z 16 ε ρ t +1 1 2 dr Finally , if b oth ρ t and ρ t +1 are smaller than 16 ε , w e use the lo w er b ound | E | · ( ρ t − ρ t +1 ) · 1 2 = | E | · Z ρ t ρ t +1 1 2 dr Summing those b ounds, w e h a v e that the num b er of edges cut b y the algorithm is at least | E | ·  Z 1 16 ε  1 − 2 r ε r − δ 2  dr + Z 16 ε 0 1 2 dr  = | E | ·  1 − 4 √ ε + 8 ε − (1 − 16 ε ) δ 2   Corollary 5 Algo rithm Recursive-S pectral-Cut is a . 53112 8 − δ appr oximat e algorithm for Max Cut. Pr oof: W rite opt = 1 − ε . If ε > 1 / 16 th en the algorithm fin ds a solution of cost > 1 / 2 and the appro ximation ratio is 16 / 30 > 5 . 333 33. If 1 / 16 ≤ ε ≤ 1 / 2, then the algorithm fin d s a s olution of cost at least 1 − 4 √ ε + 8 ε − δ / 2, and th e appro ximation ratio is at least 1 − 4 √ ε + 8 ε − δ / 2 1 − ε ≥ 1 − 4 √ ε + 8 ε 1 − ε − δ 11 If we call ρ ( ε ) := 1 − 4 √ ε +8 ε 1 − ε , then some calculus shows that, f or 1 / 16 ≤ ε ≤ 1 / 2, ρ ( ε ) is minimized at . 05496 (the smallest ro ot of − 2 x 2 + 9 x − 2 = 0) and is alwa ys at least . 531128 · · · .  5 Relati on to Cheege r’s Inequalit y In this section we compare our main r esu lt, Theorem 1 , with Cheeger’s inequalit y [ Alo86 ]. W e restrict our d iscussion to the case of regular graph . If G is a d -regular graph, A is its adjacency matrix, and M := 1 d A , then M has n eigen v alues, coun ting multiplic ities, which we sh all call λ 1 ≥ λ 2 ≥ · · · ≥ λ n . It is alw a ys the case that λ 1 = 1, and that | λ i | ≤ 1 f or eve ry i . The extremal cases are captured b y the follo wing w ell-kno wn facts: 1. λ 2 = 1 if and only if G is disconnected, that is, if and only if there is a set S , | S | ≤ | V | / 2, suc h that no edge of G lea v es S . 2. λ n = − 1 if and only if G con tains a b ip artite connected comp onent, that is, if and only if there is a set S and partition of S into disjoint sets L, R , suc h that all edges incident on S ha v e one end p oin t in L and one endp oin t in R . Cheeger’s inequalit y c haracterizes the cases in whic h λ 2 is close to 1 as those in whic h there is a set S , | S | ≤ | V | / 2 such that the num b er of ed ges b et wee n S and V − S is small compared to d | S | . If we d efine h ( G ) to b e the e dge exp ansion of G , h ( G ) = min S ⊆ V : | S |≤| V | / 2 edges ( S, V − S ) d | S | then we hav e Cheeger’s inequalit y p 2 · (1 − λ 2 ) ≥ h ( g ) ≥ 1 2 · (1 − λ 2 ) (7) Similarly , Lemmas 2 and 3 c haracterizes the cases in whic h λ n is close to − 1 as those in wh ic h there is a s et S and a partition ( L, R ) of S suc h that the num b er of edges inciden t on S wh ic h f ail to b e cut by the partition is small compared to d | S | . Define the bip artiteness r atio num b er of a graph to b e β ( G ) := min y ∈{− 1 , 0 , 1 } V P i,j | y i + y j | 2 d P i | y i | whic h is equiv ale nt to β ( G ) = min S ⊆ V , ( L,R ) partition of S 2 edges ( L ) + 2 edges ( R ) + edg es ( S, V − S ) d | S | 12 then we hav e p 2 · (1 − | λ n | ) ≥ β ( G ) ≥ 1 2 · (1 − | λ n | ) (8) There are examples in wh ic h b oth inequalities in ( 8 ) are tight within constant factors. If w e tak e an o dd cycle with n vertice s, then β ( G ) ≥ 1 n , b ecause for eve ry subset S of v ertices and for ev ery b ipartition of S there is at least one failed edge, and the num b er of edges incident on S is at m ost n . In an o dd cycle, ho wev er, d = 2 and | λ n | = 2 − O (1 /n 2 ), and so β is as large as Ω( p 1 − | λ n | ). T o see the tigh tness of the other inequalit y , start fr om a k -regular expander suc h that, sa y , max { λ 2 , | λ n |} ≤ 1 / 2. (Such graph s exist for constan t k .) Then construct G b y taking the dis- join t union of the edges of G and the edges of a k · (1 − ε ) /ε -regular b ipartite graph, so that the resulting graph is d -regular with d := k /ε . Th ere is a cut that cuts all the edges of the b ipartite graph, so β ( G ) ≤ ε , bu t the smallest eigen v alue of M is at least − 1 + k / 2 d ≥ − 1 + ε/ 2, meaning that β is O (1 − | λ n ( G ) | ). Our resu lts, as stated in ( 8 ), are not just synt actically similar to Cheeger’s inequalit y: Th ere are also similarities b etw een the pro of of Cheeger’s inequalit y and of Theorem 1 . T he analysis in Cheeger’s inequalit y relies on th e s tu dy of th e quadratic form X i,j A ( i, j ) · ( x i − x j ) 2 (9) and it is based on the in tuition that if ( 9 ) is small compared to P i x 2 i then for m ost edges ( i, j ) w e ha v e x i ≈ x j . Our analysis was based on the study of the quadratic form X i,j A ( i, j ) · ( x i + x j ) 2 (10) and the intuiti on that if ( 10 ) is sm all compared to P i x 2 i then for most edges w e ha v e x i ≈ − x j . 6 Relati on to the Go emans-Williamso n Relaxati on The dual of the Go emans-Williamson relaxation is min | E | − 1 4 P i y i sub ject to D + A − diag ( y 1 , . . . , y n )  0 (11) W e can see Lemma 2 as stating a sp ecial case of the w eak du alit y fact that the cost of eve ry feasible solution to ( 11 ) is an up p er b ound to the optimal cut in the graph. Indeed, if the optimal cut is of size > | E | · (1 − ε ), then no solution of cost ≤ | E | · (1 − ε ) can b e feasible for ( 11 ). In p articular, the solution y i = 2 εd i has cost 1 − ε and cannot b e feasible, meaning that D (1 − 2 ε ) + A cannot b e feasible, and there is a v ector x suc h that x ( D (1 − 2 ε ) + A ) x < 0. 13 In turn, Lemma 3 has the follo wing primal du al interpretat ion: giv en a graph G , there is an ε suc h that algorithm 2TSC find s L, R suc h that C + 1 2 X ≥ (1 − 2 √ ε − δ / 2) M , and th e solution y i := 2 εd i is feasible f or ( 11 ), thus sh o wing that the Max Cut optimum is at m ost 1 − ε . Giv en this pr emise, we can now view algorithm Recu rsive-Spectral -Cut as a p rimal-dual al- gorithm. A t step t of the recursion, let ρ t | E | b e the num b er of edges in the residual graph G t , and C t and X t b e the num b er of cut and cross edges in the solution L t , R t found by the algorithm. Define ε t so that 1 − ε t /ρ t is the upp er b ound on the Max Cut of G t giv en by the d ual solution asso ciated to the algorithm as ab o v e, and the algorithm satisfies C t + 1 2 X t ≥ (1 − 2 p ε t /ρ t − δ / 2) M t . Th en the du al solution at time t also prov es an upp er b ound 1 − ε t to the Max Cut optimum of G . Let ε := max t ε t ; then we ha ve (i) a du al solution pro ving that the Max Cut of G is ≤ 1 − ε , and we kno w that (ii) at ev ery step t we hav e C t + 1 2 X t ≥ (1 − 2 p ε/ρ t − δ / 2) M t . F rom fact (ii) and the analysis done in the pro of of Theorem 1 w e see the algorithm outpu ts a solution that cuts at least a 1 − 4 √ ε + 8 ε − δ / 2 f r actio n of edges, and it is able to output a f easible dual solution to the GW relaxatio n p ro ving a 1 − ε upp er b oun d to the optim um. In particular, the ratio b et w een the cost of the solution found by the algorithm and th e upp er b ound pro vided b y the dual solution is alw a ys at least . 531. 7 Quadratic Programm ing and the Max CutGain Problem Let A b e the adjacency matrix of a w eigh ted graph with no self-loops, p ossibly with negativ e w eigh ts, let d i := P j | A i,j | b e the w eigh ted degree of no d e i , and D := diag ( d 1 , . . . , d n ). Max-Cut Gain is the optimizatio n p roblem max y ∈{− 1 , 1 } V − y T Ay y T D y (12) In words, Max Cut Gain is the maxim um, o v er all cuts, of the difference b et we en the num b er of cut edges and the num b er of edges that are not cut, divided by th e total num b er of edges. Equiv alen tly , the optim um of Max Cu t Gain is ε if and only if th e optim um of Max Cut is 1 2 + 1 2 ε . (The n ame of the pr ob lem comes from the f act that one is measurin g ho w muc h one gains by using an optimum cut compared to a rand om cut, whic h only cuts a 1 / 2 fraction of edges.) Note that, u p to the scaling that w e do b y dividing b y y T D y = P i d i , w e are considering the problem max y ∈{− 1 , 1 } V y T Qy (13) where Q is an arbitrary symmetric matrix with zero es on the diagonal. Apart fr om the restriction to symm etric matrices, this is the same family of quadratic p rograms s tudied by Charik ar and Wirth [ CW04 ]. It helps in tuition, ho wev er, to con tin ue to think ab out A = − Q as the adjacency matrix of a w eigh ted und irected graph. 14 W e define the gain r atio of a graph the quantit y γ ( G ) := max y ∈{− 1 , 0 , 1 } V − y T Ay y T D y (14) In the gain r atio , w e consider all subsets S ⊆ V of v ertices, and all partitions ( L, R = S − L ) of the set S ; the ob jectiv e fun ction is the ratio b etw een t wice th e difference of cut edges minus uncut edges among the edges ind uced by S , divided b y the vo lume of S . If one imp osed the additional constrain t S = V , then one would reco ve r the Max Cu t Gain problem. Let λ n b e the smallest eigen v alue of the matrix M := D − 1 / 2 AD − 1 / 2 ; then we see that γ ( G ) ≤ | λ n | (15) b ecause | λ n | = − min z ∈ R V z T M z z T z = max x ∈ R V − x T Ax x T D x ≥ max y ∈{− 1 , 0 , 1 } V − y T Ay y T D y = γ ( G ) w e conjecture that γ ( G ) ≥ Ω | λ n | log 1 | λ n | ! (16) but we are only able to prov e the considerably w eak er result that γ ( G ) ≥ e − O (1 / | λ n | ) . W e u se the follo wing approac h. Let x ∈ R V b e a real v ector, and Y b e a d istribution ov er discrete v ectors {− 1 , 0 , 1 } V . W e s ay that Y is a ( c 1 , c 2 , δ )-go o d (randomized) rounding of x if 1. | c 1 · E Y i Y j − x i x j | ≤ δ · ( x 2 i + x 2 j ) 2. E | Y i | ≤ c 2 x 2 i W e ha ve the follo wing simple fact: Claim 6 If x is a ve c tor such that − x T Ax ≥ ε · x T D x , and Y is a a ( c 1 , c 2 , δ ) -g o o d r ounding of x , then the supp ort of Y c ontains a ve ctor y ∈ {− 1 , 0 , 1 } V such that − y T Ay ≥ 1 c 1 c 2 ( ε − 2 δ ) · y T D y Pr oof: W e hav e E X i,j − A ij Y i Y j ≥ 1 c 1   X ij − A i,j x i x j   + 1 c 1 2 δ X i d i x 2 i 15 ≥ 1 c 1 · ( ε − 2 δ ) X i d i x 2 i ≥ 1 c 1 c 2 · ( ε − 2 δ ) E X i d i | Y i | and so E P i,j − A ij Y i Y j E P i d i | Y i | ≥ 1 c 1 c 2 ( ε − 2 δ ) and in p articular there must exist a v ector y ∈ {− 1 , 0 , 1 } su c h that P i,j − A ij y i y j P i d i | y i | ≥ 1 c 1 c 2 ( ε − 2 δ )  Lemma 7 (Main) F or every x ∈ R V and every ℓ > 1 ther e is a ( c 1 , c 2 , 1 /ℓ ) -go o d r ounding of x such that c 1 · c 2 ≤ ℓ − 1 · e ℓ . Pr oof: Given x , we assu me without loss of generalit y that | x i | ≤ 1 for ev ery i , and w e consider the follo wing d istribution Y : • Pic k a threshold t ∈ [0 , 1 ] so that t 2 is un iformly distrib uted in [0 , 1]; • F or ev ery v ertex i , pairwise indep end en tly: – If | x i | > t or | x i | < t · e − ℓ , then s et Y i := 0; – If t · e − ℓ ≤ | x i | ≤ t , then set Y i := sig n ( x i ) with p robabilit y | x i | /t , and Y i := 0 with probabilit y 1 − | x i | /t . W e b egin with the calculatio n of the exp ectations E | Y i | . Claim 8 E | Y i | = 2 · ( e ℓ − 1) · x 2 i Pr oof: [Of Claim ] Th e th r eshold t is chosen according to a d istribution whose densit y function is 2 t for t ∈ [0 , 1]; conditioned on a sp ecific choi ce of t , the exp ectation of | Y i | is 0 if | x i | > t or | x i | < te − ℓ , and it is | x i | /t otherwise. Hence, we ha v e E | Y i | = Z | x i | e ℓ | x i | 2 t · | x i | t dt = Z | x i | e ℓ | x i | 2 | x i | dt = 2 · ( e ℓ − 1) · x 2 i  Claim 7 tells us that w e can tak e c 2 = 2 · ( e ℓ − 1) ≤ 2 e ℓ . The follo wing t w o claims giv e us that w e can tak e c 1 = 1 / 2 ℓ , so that c 1 c 2 ≤ 1 ℓ · e ℓ as required. 16 Claim 9 If | x i | > e ℓ | x j | , then, for every c | c E Y i Y j − x i x j | ≤ 1 ℓ x 2 i Pr oof: [Of Claim 9 ] Ju st note that, u nder the assump tion of the claim, E Y i Y j = 0, and | x i x j | ≤ e − ℓ x 2 i ≤ ℓ − 1 x 2 i .  Claim 10 If | x j | ≤ | x i | ≤ e ℓ | x j | , then     1 2 ℓ · E Y i Y j − x i x j     ≤ 1 ℓ · x 2 i Pr oof: [Of Claim 10 ] Consid er the exp ectation of Y i Y j , i 6 = j , conditioned on a fi xed choic e of t . Y i Y j = 0 wheneve r | x i | ≥ t or | x j | ≤ te − ℓ . If t is su c h that | x i | ≤ t ≤ | x j | e ℓ , then the conditional exp ectatio n of Y i Y j is x i x j /t 2 . Overal l, we h av e E Y j Y j = Z | x j | e ℓ | x i | 2 t · x i x j t 2 dt = 2 x i x j · Z | x j | e ℓ | x i | 1 t dt = 2 x i x j ·  ℓ − ln | x i | | x j |  So we hav e     1 2 ℓ · E Y i Y j − x i x j     = | x i x j | · 1 ℓ · ln | x i | | x j | = x 2 i · 1 ℓ · | x j | | x i | · ln | x i | | x j | ≤ x 2 i · 1 ℓ where the last inequalit y follo ws fr om the fact th at ρ ln 1 ρ ≤ 1 for ev ery 0 < ρ ≤ 1.  The lemma n o w follo ws.  In order to mak e the pro of constructiv e, w e need to sh ow that w e can fin d a vect or y in the sample space of Y as in the conclusion of the lemma. Supp ose that the distribution of Y describ ed ab o ve is suc h th at − E Y T AY ≥ E δ Y T D Y . A first ob s erv ation is that there must b e a thr eshold t ∗ suc h that, conditioned on that particular c hoice of t , w e still ha v e − E [ Y T AY | t = t ∗ ] ≥ δ E [ Y T D Y | t = t ∗ ]. Once w e fi nd such a threshold, w e can searc h in the sample space of Y | t = t ∗ , whic h is of p olynomial size. It remains to describ e ho w to find a threshold t ∗ as ab ov e. Let us say th at t wo th r esholds t 1 , t 2 are c ombinatorial ly indistinguishable if the sets of verti ces { i : δ t 1 ≤ | x i | ≤ t 1 } and { i : δ t 2 ≤ | x i | ≤ t 2 } are equal, and call S the set of verti ces. Then w e ha v e − E [ Y T AT | t = t 1 ] E [ Y T D Y | t = t 1 ] = − P i,j ∈ S A ij x i x i /t 2 1 P i ∈ S d i | x i | /t 1 = − 1 t 1 · P i,j ∈ S A ij x i x i P i ∈ S d i | x i | 17 and, similarly − E [ Y T AT | t = t 1 ] E [ Y T D Y | t = t 2 ] = − 1 t 2 · P i,j ∈ S A ij x i x i P i ∈ S d i | x i | so that it is alwa ys preferable to choose th e smaller threshold. This m eans that f or eve ry equiv alence class of com binatorially indistinguishable thr esh olds we only need to lo ok at on e of them, in order to fi nd t ∗ , and so we only need to consider at most 2 | V | thresholds. In particular, t ∗ can b e found in O ( | E | + | V | ) time. A nearly pairwise indep endent sample space of size ˜ O ( | V | ) can b e u s ed instead of a p erfectly pairwise in dep endent one so that the whole algorithm tak es time ˜ O ( | V | + | E | ), at the price of a o (1) additiv e loss in the approxima tion. The follo wing theorem summarizes our progress so far. Theorem 11 Ther e is a ne arly quadr atic time algorithm that in i nput a gr aph G = ( V , E ) such that γ ( G ) ≥ ε finds a set S and a p artition ( L, R ) of S whose gain is at le ast e − Ω(1 /ε ) . Pr oof: W e call the algorithm F our-Thr eshold Sp e ctr al Cu t , or 4TSC. • Algorithm 4TSC • Input: Graph G = ( V , E ) – Let A b e the adjacency matrix of G , D b e the matrix of d egrees, M := D − 1 / 2 M D − 1 / 2 . Find a v ector x ∈ R V suc h th at ε := − x T M x/x T x ≤ 2 | λ n | , where λ n is the smallest eigen v alue of M . Set ℓ = 10 /ε – F or eve ry thr eshold t in the s et { x ( i ) : i ∈ V } ∪ { e − ℓ x ( i ) : i ∈ V } ∗ Let Y 1 , . . . , Y n b e a distribution of sample space Ω t that is ε/ 10-cl ose to pairwise indep endence, and suc h that Y i ≡ 0 if | x i | > t or | x j | < e − ℓ t ; and suc h that Y i = sig n ( x i ) with p robabilit y | x i | /t otherwise. – Output the v ector y in the u nion of Ω t that maximizes P i,j A i,j | y i + y j | P i d i | y i | Using the construction of almost pairwise indep end en t random v ariables of Alon et al. [ A GHP92 ], eac h sample sp ace Ω t has size ˜ O (log n ), and can b e computed in ˜ O ( n ) time. F or eac h vec tor y , the ratio can b e computed in linear time.  By iterating th e algorithm we derive our main result of this section. Theorem 12 Ther e is a ne arly cub ic time algorithm that in input a gr aph G = ( V , E ) such that max − cut − g ain ( G ) ≥ ε finds a cut ( L, R ) of V o f gain ≥ e − Ω(1 /ε ) 8 Conclusions The motiv ating question for this w ork was to fi n d a com binatorial interpretat ion of the quantit y d − | λ n | in a d -regular graph , akin to the in terpretation of d − λ 2 pro vided b y the theory of edge expansion. 18 In establishing such an in terpretation (in terms of the quan tit y that w e call “bipartiteness ratio” in Section 5 ) we prov ed that a natural and easy-to-implemen t sp ectral algorithm p erforms n on-trivially w ell with r esp ect to the Max Cu t pr oblem. The algorithm is v ery fast in pr actic e [ OT08 ]; usin g a termination ru le that is slightl y m ore relaxed than the one us ed in this pap er (stopping when U + X > M / 2, in s tead of U + X/ 2 > M / 2), the algorithm make s at most one recursive call in all the exp eriments that w e p erformed. It wo uld b e in teresting to giv e a pro of that this is alwa ys the case. A n umb er of in tersting op en questions remain, suc h as: 1. What is the worst-ca se appr o ximation ratio of our algorithm? W e b eliev e that our b ond . 531 is not tight . 2. Is there a “purely com binatorial” alg orithm (namely , one not inv olving n umerical matrix computations) for Max Cut ac hieving an app ro ximation factor b etter than 1/2? 3. It should b e p ossible to significantly imp ro v e our b ounds for Max CutGain. Ac kno wledgeme n ts I wo uld lik e to thank an anonymo us commen ter for asking the question of the connection b et w een sp ectral tec hn iqu es and Max C ut, and Sebastian Cioaba, Sat y en Kale, James Lee and Salil V adhan for provi ding h elpful commen ts an d references to the related literature. I am grateful to Moses C harik ar for comm unicating the p ro of of Th eorem 4 , whic h substant ially impro v ed my previous analysis, and for allo wing me to pr esen t h is impro ve d analysis in this pap er. References [A CMM05] Amit Agarw al, Moses Charik ar, K onstan tin Mak aryc hev, and Y ury Mak aryc hev. O ( √ log n ) appro ximation algorithms for min UnCut, min 2CNF deletion, and d irected cut problems. In Pr o c e e dings of the 37th ACM Symp osium on The ory of Computing , pages 573–581 , 2005. 5 [A GHP92] N. Alon, O. Goldreic h, J. H ˚ astad, and R. Peralta . Simple constructions of almost k - wise indep endent random v ariables. R andom Structur es and Algorithms , 3(3): 289–304 , 1992. 18 [AK07] Sanjeev Arora and Sat y en Kale. A combinato rial, primal-dual ap p roac h to semidefinite programs. In Pr o c e e dings of the 39th ACM Symp osium on The ory of Computing , pages 227–2 36, 2007. 1 , 22 [Alo86] Noga Alon. Eigen v alues and expanders. 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In Pr o c e e dings of the 33r d ACM Symp osium on The ory of Computing , pages 453–461 , 2001. 2 , 22 21 A App endix A.1 Efficiency of the Arora-Kale Algorithm Arora and Kale [ AK07 ] describ e an algorithm for the Go emans-Williamson SDP relaxation of Max Cut whic h ac hiev es an app ro ximation r atio 1 + o (1) and runs in time ˜ O ( D max · | V | ) giv en in inpu t an unw eigh ted mult i-graph G = ( V , E ) of maxim um degree D max . 5 In particular, it is p ossible to find ( α − o (1))-appro ximate solutions to Max C ut in time ˜ O ( D max · | V | ), where α = . 878 · · · is the appro ximation ratio of the Go emans -Williamson algorithm. In this section we show that, using the Arora-Kale algorithm and a r eduction fr om [ T re01 ], it is p ossible to appro ximate Max Cut within α − o (1) in time ˜ O ( | V | + | E | ) r egardless of the degree distribution. 6 Giv en the sp arsification result discussed in Section 2 , it is sufficien t to pro ve the follo wing theorem, whic h is implicit in [ T re01 ]. Theorem 13 Ther e is a r ando mize d algorithm C and a deterministic algorithm R with the f ol low- ing pr op e rties. Given a gr aph G = ( V , E ) , algor ithm C c onstructs in ˜ O ( | V | + | E | ) time a gr aph G ′ = ( V ′ , E ′ ) of maximum de gr e e ˜ O (1) with | V ′ | = 2 | E | vertic es, such that the fol lowing happ ens with high pr ob ability: (i) maxcut ( G ′ ) ≥ maxcut ( G ) − o (1) , and (ii) given an arbitr ary solution S ′ ⊆ V ′ of c ost c in G ′ , algorithm R c onst ructs in ˜ O ( | V | + | E | ) time a solution S ⊆ V of c ost ≥ c − o (1) for G . Pr oof: W e sket c h h o w the argument in [ T re01 ] applies to Max Cu t. Define the we igh ted graph ˆ G = ( ˆ V , ˆ E ) as follo ws. (Th is graph will only b e u s ed in the analysis, and not explicitely constru cted in the r eduction.) F or eve ry v ertex v ∈ V of d egree d v , ˆ V con tains d v copies of v ; for ev ery edge ( u, v ) in E , w e hav e d u · d v edges ( ˆ u, ˆ v ) in E ′ , one for ev ery cop y ˆ u of u and for ev ery copy ˆ v of v , eac h su ch edge ha ving w eigh t 1 / ( d u · d v ). W e claim that app ro ximating Max Cu t in G is equiv alen t to appr o ximating Max Cut in ˆ G . First, it should b e clear that if ( S, V − S ) is a cut in G of cost c , then if w e define ˆ S ⊆ ˆ V to b e th e set of all copies of vertic es in S , then ( ˆ S , ˆ V − ˆ S ) is a cut of cost c in ˆ G . On the other hand, if ( ˆ S , ˆ V − ˆ S ) is a cut of cost c , then consider the distribu tion o ve r cuts in G in which a v ertex v is pic k ed to b e in S with probabilit y prop ortional to the f r actio n of copies of v whic h are in ˆ S ; the exp ected fraction of cut edges in G is exactly c , and using the metho d of conditional exp ectatio ns w e can find a cut of cost at least c in linear time. The graph G ′ is obtained b y samp ling with replace ment ˜ O ( | ˆ V | ) = ˜ O ( | V | + | E | ) edges from ˆ E , using the distribution in w hic h an edge is sampled with probabilit y p rop ortional to its w eigh t. As discussed in Section 2 , it follo ws from Chernoff b ounds that a solution of cost c in G ′ has cost c ± o (1) in ˆ G . It remains to d iscuss the complexit y of sampling G ′ : to sample one edge, we firs t p ic k a rand om edge ( u, v ) of G , and then w e pic k at random one of the copies ˆ u of u and one of the copies ˆ v 5 The Arora-Kale result is more general, but this statemen t is sufficient for our purp ose 6 The running time can b e redu ced to ˜ O ( | V | ) if th e representation of the graph is such that a random edge can b e sampled in ˜ O (1) time, and the degree of a given vertex can b e found in ˜ O (1) time. 22 of v ; this d istrib ution is equiv ale nt to r andomly sampling on e of the edges of ˆ G with probabiltiy prop ortional to its weig ht. Afte r O ( | V | + | E | ) time p repro cessing, eac h edge of G ′ can b e sampled in constan t time. 7  7 The p oint of th is discussion is that ˆ G ma y h a ve Ω( | V | 2 ) edges even if | E | = O ( | V | ), for example if th ere are tw o vertice s of degree | V | − 1. This means that it is not p ossible to explicitly construct ˆ G in ˜ O ( | V | + | E | ) time, and so one must sample edges from ˆ G without explicitly constructing ˆ G . 23