Bipartite subgraphs and the smallest eigenvalue

Transcription

1 Bipartite subgraphs and the sallest eigenvalue Noga Alon Benny Sudaov Abstract Two results dealing with the relation between the sallest eigenvalue of a graph and its bipartite subgraphs are obtained. The first result is that the sallest eigenvalue µ of any nonbipartite graph on n vertices with diaeter D and axiu degree satisfies µ + 1 (D+1)n. This iproves previous estiates and is tight up to a constant factor. The second result is the deterination of the precise approxiation guarantee of the Max Cut algorith of Goeans and Williason for graphs G = (V, E) in which the size of the ax-cut is at least A E, for all A between and 1. This extends a result of Karloff. 1 Introduction The sallest eigenvalue of (the adjacency atrix of) a graph G is closely related to properties of its bipartite subgraphs. In this paper we obtain two results based on this relation. In [8], Proble 11.9 it is proved that if G is a d-regular, non-bipartite graph on n vertices with diaeter D, then its sallest eigenvalue µ satisfies µ + d > 1 ddn. Our first result here iproves this bound as follows. Theore 1.1 Let G = (V, E) be a graph on n vertices with diaeter D, axiu degree and eigenvalues λ 1 λ... λ n. If G is non-bipartite then λ n + 1 (D + 1)n. If G = (V, E) is an undirected graph, and S is a nonepty proper subset of V, then (S, V S) denotes the cut consisting of all edges with one end in S and another one in V S. The size of the Departent of Matheatics, Rayond and Beverly Sacler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel and Institute for Advanced Study, Princeton, NJ Eail: noga@ath.tau.ac.il. Research supported in part by a USA Israeli BSF grant, by a grant fro the Israel Science Foundation and by a State of New Jersey grant. Departent of Matheatics, Rayond and Beverly Sacler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Eail: sudaov@ath.tau.ac.il. Matheatics Subject Classification (1991): 05C50, 05C85, 05C35 1

2 cut is the nuber of edges in it. The MAX CUT proble is the proble of finding a cut of axiu size in G. This is a well nown NP-hard proble (which is also MAX-SNP hard as shown in [9]- see also [6], []), and the best nown approxiation algorith for it, due to Goeans and Williason [5], is based on seidefinite prograing and an appropriate (randoized) rounding technique. It is proved in [5] that the approxiation guarantee of this algorith is at least the iniu of the function h(t)/t in (0, 1], where h(t) = 1 π arccos(1 t). This iniu is attained at t 0 = and is roughly Karloff [7] showed that this iniu is indeed the correct approxiation guarantee of the algorith, by constructing appropriate graphs. The authors of [5] also proved that their algorith has a better approxiation guarantee for graphs with large cuts. If A t 0, with t 0 as above, and the axiu cut of G = (V, E) has at least A E edges, then the expected size of the cut provided by the algorith is at least h(a) E, showing that in this case the approxiation guarantee is at least h(a)/a (which approaches 1 as A tends to 1). Here we apply the relation between the sallest eigenvalue of G and the axiu size of a cut in it to show that this result is tight for every such A, that is, the precise approxiation guarantee of the Goeans-Williason algorith for graphs in which the axiu cut is of size at least A E is h(a)/a for all A between t 0 and 1. This extends the result of Karloff, who proved the stateent for A = t 0. The technical result needed for this purpose is the following. Theore 1. For any rational η satisfying 1 < η < 0, there exists a graph H = (V, E), V = {1,..., n} and a set of unit vectors w 1,..., w n in R, 1 n, such that wi tw j = η for all i j with {i, j} E and the size of a axiu cut in H is equal to ax v i =1,v i R n 1 v t i v j = 1 w t i w j = 1 η. The rest of this short paper is organized as follows. In Section we prove Theore 1.1 and present exaples showing that its stateent is optial, up to a constant factor. In Section 3 we prove Theore 1. by constructing appropriate graphs. Our construction resebles the one in [7], but is ore general and its analysis is soewhat sipler. The ain advantage of the new construction is that unlie the one in [7], it is a Cayley graph of an abelian group and therefore its eigenvalues have a siple expression, and can be copared with each other without too uch efforts. The construction in [7] is an induced subgraph of one of our graphs. We also discuss in Section 3 the relevance of the construction to the study of the approxiation guarantee of the algorith of [5]. The final Section 4 contains soe concluding rears.

3 Non-bipartite graphs In this section we prove Theore 1.1 and show that its estiate is best possible up to a constant factor. Proof of Theore 1.1. Let A be the adjacency atrix of G = (V, E) and let V = {1,..., n}. Denote by d i the degree of the vertex i in G. Let x = (x 1,..., x n ) be an eigenvector, satisfying x = 1, corresponding to the sallest eigenvalue λ n of A. Then λ n = λ n x = x t Ax = a i,j x i x j = i,j x i x j. (1) The fact that the axiu degree of G is, together with the inequality = x i d ix i iplies that + λ n i d i x i + x i x j = (x i + x j ). Partition the vertices of the graph into two parts by taing the first part to be the set of all vertices with negative coordinates and the second one to be the set of all reaining vertices. Although it is not difficult to show, using the Perron-Frobenius theore, that both parts are nonepty, this fact is not needed in what follows. Since G is non-bipartite one of the parts ust contain an edge. Therefore there exists an edge {i, j} of G such that either both coordinates x i, x j are non-negative or both of the are negative. Without loss of generality we can assue that x 1 is positive and has axiu absolute value aong all entries of x. First consider the case that {i, j} E and x i 0, x j 0. Since the diaeter of G is D, we can assue that i D + 1, that 1,,..., i 1, i is a shortest path fro 1 to the set {i, j} and that j = i + 1. Therefore, either the path 1,..., i or the path 1,..., i, i + 1 has odd length and this length is bounded by D + 1. Let 1,..., be such a path with x 0, even and D +. Then, by the Cauchy-Schwartz inequality + λ n 1 (x i + x j ) (x i + x i+1 ) 1 1 ( 1 x i + x i ((x 1+x )+( x x 3 )+(x 3 +x 4 )+( x 4 x 5 )+...+(x 1 +x )) = 1 1 (x 1+x ) x 1 D + 1. Finally, since i x i = 1 and x 1 has axiu absolute value we conclude that + λ n x 1 D + 1 i x i n(d + 1) = 1 n(d + 1). Next consider the case that both coordinates x i, x j, {i, j} E, are negative. Using the reasoning above it follows that there exists a path 1,..., such that x < 0, is odd and D +. Thus we have 3 )

4 + λ n 1 (x i + x j ) (x i + x i+1 ) 1 1 ( 1 x i + x i ((x 1 + x ) + ( x x 3 ) + (x 3 + x 4 ) + ( x 4 x 5 ) ( x 1 x )) = 1 1 (x 1 x ) This copletes the proof. x 1 D + 1 i x i n(d + 1) = 1 n(d + 1). As a corollary we obtain the following result for d-regular graphs. Corollary.1 Let G = (V, E) be a d-regular graph on n vertices with diaeter D and eigenvalues λ 1 λ... λ n. If G is non-bipartite then λ 1 + λ n = d + λ n 1 (D + 1)n. Next we give exaples of graphs which show that the estiate of Corollary.1 (and therefore also that of Theore 1.1) is best possible up to a constant factor. In case d = consider the cycle C n of length n = D + 1. Clearly the diaeter of C D+1 is D and it is well nown (see, e.g., [8]) that its axiu eigenvalue is λ 1 = and the iniu one equals λ n = cos( Dπ D+1 ). Thus ( ) ( ) Dπ π λ 1 + λ n = + cos = 1 cos( D + 1 D + 1 ) ) π (D + 1) π Dn. To generalize the previous exaple and show that our estiate is tight even if G has diaeter D and ore vertices and/or larger degrees, we construct for d 3 a graph H D,d which is essentially a blow up of the cycle of odd length by coplete bipartite graphs. Let H D,d = (V, E) be the graph on the set of vertices V = {v i,j, u i,j 1 i D + 1, 1 j d} whose set of edges includes all edges {v i,j, u i, } for all j, except the cases j = = 1, j = = d and the edges {v i,d, v i+1,1 }, {u i,d, u i+1,1 } where i + 1 is reduced odulo D + 1. Note that by definition, the graph H D,d is d-regular, has n = (D + 1)d vertices and is non-bipartite, since it contains, for exaple, the odd cycle v 1,1, u 1,d, u,1, v,d,..., u 1,1, v 1,, u 1,, v 1,1 of length (D + 1) + 3. Thus to show that the estiate of Theore 1.1 is tight it is enough to prove the following proposition. Proposition. The graph H D,d has diaeter D + and its largest and sallest eigenvalues λ 1, λ n satisfy λ 1 + λ n = O( 1 nd ). 4

5 Proof. The diaeter is easily coputed. To prove the estiate on λ 1 + λ n, let A H be the adjacency atrix of H D,d and let λ 1 = d... λ n be the eigenvalues of A H. By the variational definition of the eigenvalues of A H (see, e.g., [10], pp ) we have that λ n = in x R n, x =1 xt A H x = in x R n, x =1 {u,v} E(H) Therefore, since H D,d is d regular, for any x R n, x = 1 we obtain λ 1 + λ n = d + λ n d v V (H) x v + {u,v} E(H) x u x v = {u,v} E(H) x u x v. () (x u + x v ). (3) We coplete the proof by constructing a particular vector x for which the last su is O( 1 nd ). Denote by y = (y 1,..., y D+1 ) an eigenvector of the cycle C D+1 corresponding to the sallest eigenvalue cos( Dπ D+1 ) and satisfying i y i = 1. Then (1) iplies that (y 1y +y y y D+1 y 1 ) = cos( Dπ D+1 ) and hence D+1 (y i + y i+1 ) + (y 1 + y D+1 ) = + cos Let x = (x v, v V (H)) be the vector defined as follows ( ) Dπ D + 1 π (D + 1). x vi,j = y i d, x ui,j = y i d i, j. Then x = i d( y i d ) = i y i = 1. Substituting the vector x into (3) and noticing that the only edges {u, v} E(H) for which x u + x v 0 are the edges of the for {v i,d, v i+1,1 } or {u i,d, u i+1,1 } we obtain that λ 1 + λ n = 1 d This copletes the proof. {u,v} E(H) (x u + x v ) = ( D+1 ( D+1 ) (y i + y i+1 ) + (y 1 + y D+1 ) ( y i + y i+1 ) + ( y ) 1 + y D+1 ) d d π d(d + 1) π Dn. 3 Max Cut 3.1 The Goeans-Williason algorith and its perforance We first describe the algorith of Goeans and Williason. For siplicity, we consider the unweighted case. More details appear in [5]. The MAX CUT proble is that of finding a cut (S, V S) of axiu size in a given input graph G = (V, E), V = {1,..., n}. By assigning a variable x i = +1 to each vertex i in S and 5

6 x i = 1 to each vertex i in V S, it follows that this is equivalent to axiizing the value of 1 x i x j, over all x i { 1, 1}. This proble is well nown to be NP-hard, but one can relax it to the polynoially-solvable proble of finding the axiu ax v i =1 1 vi tv j, where each v i ranges over all n-diensional unit vectors. Note that all our vectors are considered as colun vectors and hence v t u is siply the inner product of v and u. This is a seidefinite prograing proble which can be solved (up to an exponentially sall additive error) in polynoial tie. The last expression is a relaxation of the ax cut proble, since the vectors v i = (x i, 0,..., 0) for a feasible solution of the seidefinite progra. Therefore, the optial value z of this progra is at least as large as the size of the ax cut of G, which we denote by OP T (G). Given a solution v 1,..., v n of the seidefinite progra, Goeans and Williason suggested the following rounding procedure. Choose a rando unit vector r and define S = {i r t v i 0} and V S = {i r t v i > 0}. This supplies a cut (S, V S) of the graph G. Let W denote the size of the rando cut produced in this way and let E[W ] be its expectation. By linearity of expectation, the expected size is the su, over all {i, j} E, of the probabilities that the vertices i and j lie in opposite sides of the cut. This last probability is precisely arccos(v t i v j)/π. Thus the expected value of the weight of the rando cut is exactly arccos(vi tv j). π However the optial value z of the seidefinite progra is equal to z = 1 vi tv j. Therefore the ratio between E[W ] and the optial value z satisfies E[W ] z = arccos(vt i v j)/π arccos(vi t (1 vt i v in v j)/π j)/ (1 vi tv j)/. Denote α = π in θ 0<θ π 1 cos θ. An easy coputation gives that the iniu α is attained at θ = , the nonzero root of cos θ +θ sin θ = 1, and that α ( , ). Thus, E[W ] α z, and since the value of z is at least as large as the weight OP T of the axiu cut, we conclude that E[W ] α OP T. It follows that the Goeans-Williason algorith supplies an α-approxiation for MAX CUT. Moreover, by the above discussion, the expected size of the cut produced by the algorith is not better than αop T in case OP T = z and for an optial solution v 1,..., v n for the seidefinite prograing proble, arccos(vt i v j)/π (1 v t i v j)/ = α for all {i, j} E. 6

7 In case the value of the seidefinite progra is a large fraction of the total nuber of edges of G, the above reasoning together with a siple convexity arguent is used in [5] to show that the perforance of the algorith is better. Put h(t) = arccos(1 t)/π and let t 0 be the value of t for which h(t)/t attains its iniu in the interval (0, 1]. Then t 0 is approxiately Define A = z / E. If A t 0 then, as shown in [5], E[W ] h(a) A z h(a) A OP T. Here, as before, the actual OP T in case OP T = z expected size of the cut produced by the algorith is not better than h(a) A and for an optial solution v 1,..., v n for the seidefinite prograing proble, vi tv j = 1 A for all {i, j} E. Karloff [7] proved that the approxiation ratio of the algorith is exactly α. To do so, he constructed, for any sall δ > 0, a graph G and vectors v 1,..., v n which for an optial solution of the seidefinite progra whose value is z (G), such that OP T (G) = z (G) and arccos(vt i v j)/π (1 vi tv < α + δ j)/ for all {i, j} E. Theore 1. is a generalization of his result, and shows that the analysis of Goeans and Williason is tight not only for the worst case, but also for graphs in which the size of the axiu cut is a larger fraction of the nuber of edges. Applying the above analysis to the graph H fro Theore 1. together with the vectors w i as the solution of the seidefinite progra we obtain that in this case A = 1 η and the approxiation ratio is precisely E[W ] z (H) = E[W ] OP T (H) = in arccos(wi tw j)/π (H) (1 wi tw j)/ = arccos η π 1 η = h(a) A. 3. The proof of Theore 1. Our construction is based on the properties of graphs arising fro the Haing Association Schee over the binary alphabet. Let V = {v 1,..., v n }, n = be the set of all vectors of length over the alphabet { 1, +1}. For any two vectors x, y V denote by d(x, y) their Haing distance, that is, the nuber of coordinates in which they differ. The Haing graph H = H(,, b) is the graph whose vertex set is V in which two vertices x, y V are adjacent if and only if d(x, y) = b. Here we consider only even values of b which are greater than /. We show that for any rational η there exists an appropriate Haing graph which satisfies the assertion of Theore 1.. We ay and will assue, whenever this is needed, that is sufficiently large. Note that by definition, H(,, b) is a Cayley graph of the ultiplicative group Z = { 1, +1} with respect to the set U of all vectors with exactly b coordinates equal to 1. Therefore (see, e.g., [8], Proble 11.8 and the hint to its solution) the eigenvectors of H(,, b) are the ultiplicative characters χ I of Z, where χ I(x) = i I x i, I ranges over all subsets of {1,..., }. The eigenvalue corresponding to χ I is x U χ I(x). The eigenvalues of H are thus equal to the so called binary 7

8 Krawtchou polynoials (see [3]), ( )( ) Pb () = ( 1) j, 0. (4) j b j j=0 The eigenvalue P b () corresponds to the characters χ I with I = and thus has ultiplicity ( ). Consider any two adjacent vertices of H(,, b), v i and v j. By the definition of H, the inner product vi tv j is b. Choose and b such that b > / is even and b possible since η is a rational nuber, 1 < η < 0. Let w i = 1 = η. This is always v i for all i; thus w i = 1 and w t i w j = η for any pair of adjacent vertices. We clai that for such choice of and b the Haing graph H(,, b) together with the set of vectors w i, 1 i n, satisfy the assertion of Theore 1.. To prove this we first need to establish a connection between the sallest eigenvalue of a graph and the seidefinite relaxation of the ax cut proble. Proposition 3.1 Let G = (V, E) be a graph on the set V = {1,,..., n} of n vertices, with adjacency atrix A = (a ij ), and let λ 1... λ n be the eigenvalues of A. Then i<j a ij 1 v t i v j 1 E 1 4 λ n n, for any > 0 and any set v 1,..., v n of unit vectors in R. Proof. Let B = (b ij ) be the n atrix whose rows are the vectors v t 1,..., vt n. Denote by u 1,..., u the coluns of B. By definition we have u i = ij b ij = n v i = n. Therefore i<j a ij 1 v t i v j = 1 E 1 a ij v iv t j = 1 E 1 u t 4 iau i i<j By the variational definition of the eigenvalues of A (see equation ()), for any vector z R n, z t Az λ n z and equality holds if and only if Az = λ n z. This iplies that i<j a ij 1 v t i v j 1 E 1 4 λ n u i = 1 E 1 4 λ n n. Note that in the last expression equality holds if and only if each u i is an eigenvector of A with eigenvalue λ n. As we show later, the sallest eigenvalue of the adjacency atrix A H = (a ij ) of the graph H(,, b) is P b (1). By the above discussion it has ultiplicity ( 1 ) = and eigenvectors u1,..., u with ±1 coordinates, where for each vertex v j = (v j1,..., v j ), u i (v j ) = v ji. Therefore, the coluns of 8

9 the atrix, whose rows are the vectors w i, are the eigenvectors 1 u i of A H corresponding to the eigenvalue P b (1). By the last sentence in the proof of Proposition 3.1 it follows that ax v l =1, l i<j a ij 1 v t i v j = 1 E(H) 1 4 P b (1) n = i<j 1 wi t a w j ij. On the other hand, u i is a vector with ±1 coordinates. Thus the coordinates of u i correspond to a cut in H(,, b) of size equal to <j a j 1 u i (v )u i (v j ) = 1 E(H) 1 4 ut ia H u i = 1 E(H) 1 4 P b (1) u i = 1 E(H) 1 4 P b (1) n. Thus the size of a axiu cut in H(,, b) is equal to the optial value of the seidefinite progra (see Proposition 3.1). following stateent. To coplete the proof of Theore 1. it reains to prove the Proposition 3. Let Pb (), 0 be the binary Krawtchou polynoials and let b be an even integer satisfying b = 1 η for soe fixed 1 < η < 0. Then P b (1) P b () for all 0, > 0 (η). Proof. We need the following well nown properties of the Krawtchou polynoials (see, e.g., [3]), ( )P b ( + 1) = ( b)p b () P b ( 1), (5) P b () = ( 1) b P b ( ), P b () = ( ) ( ) 1 P (b). (6) b By (4) we have that Pb ( b (1) = ) b < 0 < P b (0) = ( ) b and P b () = Pb ( ) since b is even. Therefore it is enough to prove the stateent of the proposition only for 1 /. First assue that is at ost 1+η. Then the equality (5) iplies that Pb ( + 1) = 1 ( b)p b () Pb ( 1) b P b () + P b ( 1) b + ax( P b (), P b ( 1) ). Since b is equal to 1 η 1+η b + and, it follows that = b 1 1. Therefore, arguing by induction on we obtain that Pb ( + 1) ax( P b (1), P b () ). By calculation fro (4), Pb () = ( b) ( ) ( 1) b < P b (1). This proves that Pb (1) P b 1+η () for all. Fro now on, till the end of the proof, c 1, c, c 3,... always denote positive constants depending only on η. Whenever needed we use the assuption that is sufficiently large as a function of η. 9

10 To coplete the proof we show, next, that for 1+η / the value of P b () is at ost c 1 1/3( ) b. This would iply that P b (1) = b ( ) b P b (), since by our assuptions about b, b where = η is a constant, bounded away fro zero. By the equality (6) Pb ( () ) = P ( (b) ) = b S 1 = ( b b ( 1) j j j=0 r j q ( 1) j ( b j )( b j )( b j )( )( ) 1 = S 1 + S, S contains all the reaining suands, and r = (b/ 1/3 ) + c as well as q = (b/ + 1/3 ) + c 3 are chosen so that S 1 contains an even nuber of ters. Note that S 1 is a su of at ost c 4 /3 suands t j = ( 1) j( ) 1 ( ( b)( b ) ( j j b )( b j+1 j 1) ), where j runs over all integers equal to r odulo in an appropriate interval. Therefore to bound the absolute value S 1 of S 1 it ) 1, is enough to bound t j for r j q. A siple calculation shows that t j = ( ) 1 ( )( b b j + 1 j 1 ) (j b) + ( b + j + 1) (b j)( j) Fro the assuption about j we have that j b c 5 /3 c 5 5/3 and that b j > j (1 b/ 1/3 ) c 6 /c 7 /c 8. Thus t j c 9 1/3( ) 1 ( b )( b j+1 j 1). Hence we obtain the following inequality, ( ) q 1 S 1 t j c ( )( ) b b 3 = c 9 1 3, j + 1 j 1 j=r j= where here we used the fact that + ( b )( b ( j= j+1 j 1) = ). Next we obtain an upper bound on S. Let t = 1/3 and p = b/; by definition S (p t) j=1 ( ) 1 ( b j )( b j ) + j=(p+t) ( ) 1 ( b j )( b j Note that the right hand side in the last inequality is exactly the probability that a hypergeoetric distribution with paraeters (, b, ) deviates by t fro its expectation. By the result of [1], the probability of this event is bounded by e t e 1/3 /c 10. This iplies that S c 11 1/3. Therefore P b () ( n b) S 1 + S c b = P b (1) ( b ) It follows that Pb (1) P b () for all 1 /. Since the value of P b (1) is negative, Pb (1) = P b (1) P b () P b (). This copletes the proof of the proposition. 10. )..

11 4 Concluding rears 1. The exaples described in Section can be easily extended to siilar exaples with other values of the degree d, the diaeter D and the nuber of vertices n, by replacing the coplete bipartite graphs substituted in these exaples with other regular bipartite graphs.. In [8], Proble 11.9 it is shown that the difference between the first and second largest eigenvalues of d-regular graphs with n vertices and diaeter D is always bigger than 1 Dn. This is essentially tight, as the exaples in Section also show that there are d-regular graphs with n vertices and diaeter D for which the difference between the largest and second largest eigenvalues is O( 1 Dn ). Indeed, consider the graph H = H D,d. Denote by y = (y 1,..., y D+1 ) an eigenvector of the cycle C D+1 corresponding to the eigenvalue cos( π D+1 ) and satisfying i y i = 1. Let x = (x v, v V (H)) be the vector such that x vi,j = x ui,j = y i d. By definition x satisfies v V (H) x v = d i y i = 0, x = 1, and a coputation siilar to the one in Section shows that the second eigenvalue of H satisfies λ (H) x t A H x d O( 1 Dn ). This iplies that λ 1 (H) λ (H) = d λ (H) = O( 1 Dn ). It is worth noting that the hidden constant in the O( 1 Dn ) bound above can be iproved by a factor of two in several ways, for exaple, by replacing the coplete bipartite graphs with coplete graphs in the construction of the graph H. 3. Let a ij, 1 i < j n, and b be reals. We call a constraint a ij (viv t j ) b i<j valid if it is satisfied whenever each v i is an integer in { 1, 1}. Feige, Goeans and Williason (see [4],[5]) proposed adding to the seidefinite progra a faily of valid constrains, in the hope of narrowing the gap between the optial value of the seidefinite progra and the weight of the ax cut. It is easy to see that, as observed in [7], since the vectors w 1,..., w n fro Section 3 have all their coordinates equal to ±1/ they satisfy any valid constraint. Therefore the proof of Theore 1. shows that the addition of any faily of valid constraints cannot iprove the perforance ratio of the Goeans-Williason algorith even for graphs containing large cuts. 4. Very recently we deterined, together with U. Zwic, the precise approxiation guarantee of the Goeans Williason algorith for graphs G = (V, E) in which the size of the ax cut is at least A E for all values of A ( 1/). It turns out that this approxiation guarantee is h(t 0 )/t 0 for all A between 1/ and t 0, where t 0 is as in Section 1 (note that for A close to 1/ 11

12 this gives a cut of size less than half the nuber of edges!). The exaples deonstrating this fact are based on the ones constructed here, but require several additional ideas. The details will appear soewhere else. Acnowledgent. We would lie to than M. Krivelevich for his coents on an early version of the paper. We are also grateful to an anonyous referee for his useful coents. References [1] V. Chvátal, The tail of the hypergeoetric distribution, Discrete Math. 5 (1979), [] P. Crescenzi, R. Silvestri and L. Trevisan, To Weight or not to Weight: Where is the Question? Proc. of the 4th Israeli Syposiu on Theory of Coputing and Systes (ISTCS 96), IEEE (1996), [3] G. Cohen, I. Honala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland, Asterda, [4] U. Feige and M. Goeans, Approxiating the value of two prover proof systes, with applications to MAX SAT and MAX DICUT, Proc. Third Israel Syposiu on Theory of Coputing and Systes, Israel, [5] M. Goeans and D. Williason, Iproved approxiation algoriths for axiu cut and satisfiability probles using seidefinite prograing, J. ACM 4 (1995), [6] J. Håstad, Soe optial inapproxiability results, Proc. of the 9 th ACM STOC, ACM Press (1997), (Full version available as E-CCC Report nuber TR ) [7] H. Karloff, How good is the Goeans-Williason MAX CUT algorith?, Proc. of the 8 th ACM STOC, ACM Press (1996), [8] L. Lovász, Cobinatorial Probles and Exercises, North-Holland, Asterda, [9] C.H. Papadiitriou and M. Yannaais, Optiization, Approxiation, and Coplexity Classes, JCSS 43 (1991), [10] J. H. Wilinson, The Algebraic Eigenvalue Proble, Clarendon Press,