. A central drawback of primal-dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order. We present a method that allows to compute acceptable approximations to the optimal solution of large problems within reasonable time. Semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems. These are convex nonsmooth programming problems and can be solved by bundle methods. We propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored for eigenvalue problems. Convergence follows from the traditional approach but a proof is included for completene...

We compare several semidefinite relaxations for the cut polytope obtained by applying the lift and project methods of Lov'asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the max-cut problem. This relaxation Q t (G) can be defined as the projection on the edge subspace of the set F t (n), which consists of the matrices indexed by all subsets of [1; n] of cardinality t + 1 with the same parity as t + 1 and having the property that their (I ; J)-th entry depends only on the symmetric difference of the sets I and J . The set F 0 (n) is the basic semidefinite relaxation of max-cut consisting of the semidefinite matrices of order n with an all ones diagonal, while Fn\Gamma2 (n) is the 2 n\Gamma1 -dimensional simplex with the cut matrices as vertices. We show the following geometric properties: If Y 2 F t (n) has rank t + 1, then Y can be written as a convex combination of at most 2 t cut matrices, extending a result of Anjos and Wolkowicz for the case t = 1; any 2 t+1 cut matrices form a face of F t (n) for t = 0; 1; n \Gamma 2. The class L t of the graphs G for which Q t (G) is the cut polytope of G is shown to be closed under taking minors. The graph K 7 is a forbidden minor for membership in L 2 , while K 3 and K 5 are the only minimal forbidden minors for the classes L 0 and L 1 , respectively. 1

Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length 6 d, and (ii) arbitrary real-valued non-decreasing functions on 6 d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, non-trivial lower bounds for circuits computing explicit functions even when d !1. The proof is relatively simple and direct, and combines the bottlenecks counting method of Haken with the idea of finite limit due to Sipser. We demonstrate the criterion by super-polynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial t-designs. We then derive exponential lower bounds for clique-like graph functions of Tardos, thus establishing an exponential gap between the monotone real and non-monotone Boolean circuit complexities. Since we allow real gates, the criterion...

In this paper we study the lower bounds problem for monotone circuits. The main goal is to extend and simplify the well known method of approximations proposed by A. Razborov in 1985. The main result is the following combinatorial criterion for the monotone circuit complexity: a monotone Boolean function f(X) of n variables X = fx 1 ; : : : ; x n g requires monotone circuits of size exp(\Omega\Gamma t= log t)) if there is a family F ` 2 X such that: (i) each set in F is either a minterm or a maxterm of f; and (ii) D k (F)=D k+1 (F) t for every k = 0; 1; : : : ; t \Gamma 1: Here D k (F) is the k-th degree of F , i.e. maximum cardinality of a subfamily H ` F with j " Hj k: 1 Introduction The question of determining how much economy the universal non-monotone basis f; ; :g provides over the monotone basis f; g has been a long standing open problem in Boolean circuit complexity. In 1985, Razborov [10, 11] achieved a major development in this direction. He worked out the, so-called,...

The semidefinite programming (SDP) formulation of the Lovász theta number ( G) does not give only one of the best polynomial bounds on the chromatic number of a graph (G) but also leads to a heuristic for the graph coloring problem. Karger, Motwani and Sudan have proved the best known worst case bound for the number of the colors needed by such a heuristic. Their ideas can be employed to produce a recursive heuristic based on the Lovász theta number or its strengthenings. Those recursive heuristics can benefit from the standard combinatorial tricks for the graph coloring.

In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lov'asz and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a linear function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semidefinite programming relaxation of Lov'asz and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof. Tetsuya Fujie Department of Management Science, Kobe University of Commerce, Kobe 651-2197, Japan. Phone : +81-78-794-6161, FAX :...