The maximal stable set problem : Copositive programming and Semidefinite Relaxations Kartik Krishnan Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY 12180 USA kartis@rpi.edu http://www.rpi.edu/ kartis Weekly Optimization Seminar 26th October 2001
Copositive programming 1 Contents 1. The maximal stable set problem 2. Conic programming (Linear, Semidefinite and Copositive Programming) 3. Rewriting maximal stable set as a copositive program 4. Approximating the copositive cone using LMI s (Parrilo) 5. Series of SDP and LP relaxations for the maximal stable set problem (De Klerk and Pasechnik) 6. Finiteness of the procedure (George Polya) 7. Some computational results on the 5 cycle problem
Copositive programming 2 1 The maximal stable set problem Given a graph G = (V, E). A subset V V is a stable set of G if the induced subgraph on V contains no edges. Max Stable Set : Find the stable set of maximum cardinality. Max Stable Set is equivalent to Max Clique in Ḡ (complementary graph). Hard to approximate within V 1 ǫ (Hastad 1999). Best known approximation guarantee : O( (Boppana 1992). V ) (log V ) 2
Copositive programming 3 2 Conic Programming 1. A set K is a cone if x, y K λ(x + y) K, λ 0. 2. For a cone K, the dual cone K is the set K = {y :< x, y > 0, x K} 3. Some cones of matrices The n n symmetric matrices : S n = {X R n2, X = X T } Positive semidefinite cone : S n + = {X S n, y T Xy 0, y R n } Copositive cone : C n = {X S n, y T Xy 0, y 0} Completely positive cone : Cn = {X = k i=1 y i yi T, y i R n, y i 0} Nonnegative matrices : N n = {X S n, X ij 0}
Copositive programming 4 3 Conic Programming (continued) Given a cone K and dual cone K. Conic Programming min X K Tr(CX), Tr(A i X) = b i, i = 1,..., m Dual Problem max y R m b T y, m i=1 y i A i + S = C, S K If K = S + n - Semidefinite Programming (Self Dual) If K = N n - Linear Programming (Self Dual) If K = C n - Copositive Programming
Copositive programming 5 4 Complexity of conic programming If the cone K and its dual K have an easily computable self concordant barrier function, then can use the interior point machinery to develop polynomial time algorithms (Nesterov and Nemirovskii) This is true for linear (LP) and semidefinite programming (SDP) For the copositive cone C n and its dual Cn there is no computable self concordant barrier function. We have a convex problem, that is hard to solve!. Deciding non-membership of C n is NP Complete (Kabadi and Murthy 1987) No finite procedure is known to test membership of C n Approximate the copositive cone by linear matrix inequalities (LMI s) (Parrilo)
Copositive programming 6 5 The Lovasz θ function subject to θ(g) = max Tr(ee T X) Let X ij = 0 {i, j} E Tr(X) = 1 X S n + α(g) denote the size of maximum stable set in G χ(ḡ) denote the chromatic number of Ḡ Lovasz s Sandwich Theorem α(g) θ(g) χ(ḡ) For the 5-cycle θ(g) = 5
Copositive programming 7 6 Max Stable set Copositive program Here A is the weighted adjacency matrix. f(x) = max x 0, ni=1 x i =1 {i,j} E x i x j ( 1 2 xt Ax) 1. Let x be any distribution of vertex weights (x 0). v k, s k is the sum of weights of all vertices adjacent to v k. Consider two non adjacent vertices v i and v j. Assume s i s j. 2. Move the weight x j from v j to v i. The new weight of v i is x i + x j, and that of v j is now zero. For this new distribution x, we have f(x ) = f(x) + x j s i x j s j f(x) Repeat this procedure. 3. Thus we have an optimal distribution where all the weights are concentrated on a max clique. For two vertices on this clique with x i x j, choose an ǫ with 0 < ǫ < (x i x j ), and change x i and x j to x i ǫ and x i + ǫ. The new distribution satisfies f(x ) = f(x) + ǫ(x i x j ) ǫ 2 > f(x) Thus the maximal value is attained for x i = 1 k on a max clique, and x i = 0 elsewhere.
Copositive programming 8 7 Max Stable Set Copositive program 1. Since a max clique of size k has k(k 1) 2 edges, we have f(x) = max {x 0,e T x=1} 1 2 xt Ax = 1 2 (1 1 k ) 2. Since the max clique on G is the max stable set on Ḡ, and (ee T A) = (A + I), where A now reads as the adjacency matrix of the complementary graph, we have 1 α(g) = min {x 0,e T x=1} xt (A + I)x Chapter 27 on Turan s graph theorem, Proofs from the book 3. Minimising quadratic function over simplex copositive program problem (De Klerk et al) α(g) = max{tr(ee T X) : Tr((A + I)X) = 1, X C n } with dual α(g) = min λ R {λ : λ(i + A) ee T C n } (Strong duality makes the two problems equivalent!)
Copositive programming 9 8 SDP Tests for copositivity 1. A sum of squares decomposition nonnegativity of the quadratic form 2. Copositivity requirement for M S n : P(x) = n i,j=1 M ij x 2 ix 2 j 0, x R n 3. Rewrite P(x) as x T M x where x = [x 2 1,..., x2 n, x 1x 2, x 1 x 3,..., x n 1 x n ] T M is of order n + 1 n(n 1) 2 4. A necessary condition for a sum of squares decomposition is M S + n. This defines the semidefinite cone K 0 n. 5. A weaker condition is that M Nn. This defines the linear programming cone C 0 n. We have C0 n K0 n. 6. Kn 0 and C0 n serve as preliminary approximations to C n, and optimization over them leads to an SDP and LP respectively.
Copositive programming 10 9 Higher order sufficient conditions Does P(x)( n i=1 x 2 i) r = ( n i,j=1 M ij x 2 ix 2 j)( n i=1 x 2 i) r allow for an SOS decomposition? These generate tighter sufficient conditions for copositivity. An idea due to Pablo Parrilo. r=0 Iff M S n + + N n r=1 Iff M satisfies M M i S n +, i = 1,..., n Mii i = 0, i = 1,..., n Mjj i + Mi ij + Mj ij = 0, i j Mjk i + Mj ik + Mij k 0, i j k where M i S n, i = 1,..., n. Other r Iff M K r C n
Copositive programming 11 10 Finite termination : Polya s theorem We have K 0 K 1... K r C n Can we find an r so we that we have the entire copositive cone C n? Answer is yes!. We need a theorem due to George Polya. Theorem 1 Let f be a homogeneous polynomial positive on the simplex = {z R n : n i=1 z i = 1, z 0} For sufficiently large N all the coefficients of ( n i=1 z i ) N f(z) are positive. Somes estimates on N can be found in Powers and Reznick.
Copositive programming 12 11 Finite termination : Polya s theorem Some comments on Polya s theorem Set f(z) = z T Mz in Polya s theorem, and let z i = x 2 i 1. Remember that Cn r is the cone of matrices for which P r (x) = x T M x has nonnegative coefficients, i.e. M N n. 2. Polya s theorem implies that some C N n, for N large contains the copositive cone C n. 3. Since we have Cn r Kr n, some KN n with N < N also contains the copositive cone C n. 4. We can thus approximate the copositive cone C n by an LP and SDP (exactly!). 5. In a worst case, these LP and SDP relaxations can be exponential in the size of the original copositive program.
Copositive programming 13 12 Successive approximations of α(g) Relax the copositive program α(g) = min λ {λ : λ(i + A) ee T C n } Define the following approximations θ r (G) = min λ {λ : λ(i + A) ee T K r, r = 0, 1,...} Finally a theorem due to De Klerk and Pasechnik Theorem 2 θ 0 (G) θ 1 (G)... θ N (G) = α(g) for N α(g) 2 From a numerical point of view, we can take the floor of θ N (G) for N sufficiently large, to estimate α(g).
Copositive programming 14 13 Results for the 5 cycle 1. α(g) = 2, θ(g) = θ 0 (G) = 5. 2. However θ 1 (G) = min λ s.t. λ(i + A) ee T M i S n +, i = 1,..., n Mii i = 0, i = 1,..., n M i jj + 2Mj ij = 0, i j M i jk + M j ik + M k ij 0, i < j < k = 2 (Hooray!) (I used SDPT3-2.2 to verify this) 3. The SDP has n +n(n 1)+ ( ) n 3 constraints, and the solution matrix S n 2. It is a block diagonal matrix with n blocks.
Copositive programming 15 References [1] E. De Klerk, Approximating the stability number of a graph via copositive programming, Talk given at the University of Vienna, November 13th 2000. [2] E. De Klerk and D. Pasechnik, Approximating the stability number of a graph via copositive programming, To appear in the SIAM Journal on Optimization, January 2001. [3] Pablo Parrilo, SDP tests for higher order quadratic programming relaxations and matrix copositivity, 7th DIMACS Implementation Challenge, Rutgers University, October 2000. [4] M. Aigner and G.M. Ziegler, Proofs from the book, Springer, 1999.