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A Spectral Bundle Method for Semidefinite Programming
 SIAM JOURNAL ON OPTIMIZATION
, 1997
"... A central drawback of primaldual 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 applica ..."
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Cited by 173 (8 self)
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A central drawback of primaldual 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...
Semidefinite Relaxations for MaxCut
 The Sharpest Cut, Festschrift in Honor of M. Padberg's 60th Birthday. SIAM
, 2001
"... 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 maxcut problem. Th ..."
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Cited by 11 (2 self)
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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 maxcut 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 maxcut 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
Combinatorics of Monotone Computations
 Combinatorica
, 1998
"... 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 realvalued nondecreasing functions on 6 d variables. This r ..."
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Cited by 8 (4 self)
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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 realvalued nondecreasing functions on 6 d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, nontrivial 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 superpolynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial tdesigns. We then derive exponential lower bounds for cliquelike graph functions of Tardos, thus establishing an exponential gap between the monotone real and nonmonotone Boolean circuit complexities. Since we allow real gates, the criterion...
A Criterion for Monotone Circuit Complexity
, 1991
"... 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 fun ..."
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Cited by 5 (2 self)
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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 kth 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 nonmonotone 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, socalled,...
Combinatorial Tricks and Lovász Theta Function Applied to the Graph Coloring Problem
"... 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 bo ..."
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Cited by 3 (0 self)
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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.
Some remarks on the Shannon capacity of odd cycles
"... We tackle the problem of estimating the Shannon capacity of cycles of odd length. We present some strategies which allow us to find tight bounds on the Shannon capacity of cycles of various odd lengths, and suggest that the difficulty of obtaining a general result may be related to different beh ..."
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Cited by 2 (0 self)
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We tackle the problem of estimating the Shannon capacity of cycles of odd length. We present some strategies which allow us to find tight bounds on the Shannon capacity of cycles of various odd lengths, and suggest that the difficulty of obtaining a general result may be related to different behaviours of the capacity, depending on the "structure" of the odd integer representing the cycle length. We also describe the outcomes of some experiments, from which we derive the evidence that the Shannon capacity of odd cycles is extremely close to the value of the Lov'asz theta function.
A Semidefinite Programming Relaxation for the Generalized Stable Set Problem
 KOBE UNIVERSITY OF COMMERCE
, 2000
"... 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 ove ..."
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Cited by 1 (1 self)
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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.