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38
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. This re ..."
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Cited by 8 (1 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
Semidefinite bounds for the stability number of a graph via sums of squares of polynomials
, 2007
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Semidefinite optimization approaches for satisfiability and maximumsatisfiability problems
 J. Satisf. Bool. Model. Comput
"... Semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite programming has had a truly significant impact. This paper surveys some of the res ..."
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Cited by 7 (3 self)
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Semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite programming has had a truly significant impact. This paper surveys some of the results obtained in the application of semidefinite programming to satisfiability and maximumsatisfiability problems. The approaches presented in some detail include the groundbreaking approximation algorithm of Goemans and Williamson for MAX2SAT, the Gap relaxation of de Klerk, van Maaren and Warners, and strengthenings of the Gap relaxation based on the Lasserre hierarchy of semidefinite liftings for polynomial optimization problems. We include theoretical and computational comparisons of the aforementioned semidefinite relaxations for the special case of 3SAT, and conclude with a review of the most recent results in the application of semidefinite programming to SAT and MAXSAT.
Fixed linear crossing minimization by reduction to the maximum cut problem
 in Proc 12th Ann. Int. Computing and Combinatorics Conference (COCOON’06
"... Abstract. Many reallife scheduling, routing and location problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear ..."
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Cited by 6 (0 self)
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Abstract. Many reallife scheduling, routing and location problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the socalled fixed linear crossing number problem (FLCNP). We show that this N Phard problem can be reduced to the wellknown maximum cut problem. The latter problem was intensively studied in the literature; efficient exact algorithms based on the branchandcut technique have been developed. By an experimental evaluation on a variety of graphs, we show that using this reduction for solving FLCNP compares favorably to earlier branchandbound algorithms. 1
COMPUTING SEMIDEFINITE PROGRAMMING LOWER BOUNDS FOR THE (FRACTIONAL) CHROMATIC NUMBER VIA . . .
 SIAM J. OPTIM. VOL. 19, NO. 2, PP. 592–615
, 2008
"... Recently we investigated in [SIAM J. Optim., 19 (2008), pp. 572–591] hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. In particular, we introduced two hierarchies of lower bounds: the “ψ”hierarchy converging to the fractional chromatic number and the “Ψ”hierar ..."
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Cited by 6 (1 self)
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Recently we investigated in [SIAM J. Optim., 19 (2008), pp. 572–591] hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. In particular, we introduced two hierarchies of lower bounds: the “ψ”hierarchy converging to the fractional chromatic number and the “Ψ”hierarchy converging to the chromatic number of a graph. In both hierarchies the first order bounds are related to the Lovász theta number, while the second order bounds would already be too costly to compute for large graphs. As an alternative, relaxations of the second order bounds are proposed in [SIAM J. Optim., 19 (2008), pp. 572–591]. We present here our experimental results with these relaxed bounds for Hamming graphs, Kneser graphs, and DIMACS benchmark graphs. Symmetry reduction plays a crucial role as it permits us to compute the bounds by using more compact semidefinite programs. In particular, for Hamming and Kneser graphs, we use the explicit blockdiagonalization of the Terwilliger algebra given by Schrijver [IEEE Trans. Inform. Theory, 51 (2005), pp. 2859–2866]. Our numerical results indicate that the new bounds can be much stronger than the Lovász theta number. For some of the DIMACS instances we improve the best known lower bounds significantly.
Blockdiagonal semidefinite programming hierarchies for 0/1 programming
 OPERATIONS RESEARCH LETTERS
, 2009
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Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert’s Nullstellensatz
, 2009
"... Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g.,a graph is 3colourable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. For an infeasible p ..."
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Cited by 4 (2 self)
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Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g.,a graph is 3colourable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows. In the first part of the paper, we show that the minimum degree of a Nullstellensatz certificate for the nonexistence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non3colourability, we proved that the minimum degree of a Nullstellensatz certificate is at least four. Our efforts so far have only yielded graphs with Nullstellensatz certificates of precisely that degree. In the second part of this paper, for the purpose of computation, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph,
Copositive optimization – recent developments and applications
 European Journal of Operational Research
, 2012
"... Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear const ..."
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Cited by 4 (1 self)
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Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of optimization is impressive, since problem classes both in the continuous and discrete world, as well as both deterministic and stochastic models are covered. Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NPhard combinatorial optimization problems. Here some of the recent success stories are told, along with principles, algorithms and applications. 1.
A Convex Optimisation Framework for the UnequalAreas Facility Layout Problem
, 2007
"... facility layout; semidefinite programming; convex programming; global optimisation. ..."
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Cited by 3 (0 self)
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facility layout; semidefinite programming; convex programming; global optimisation.
Global lower bounds for the VLSI macrocell floorplanning problem using semidefinite optimization
 In Proceedings of IWSOC 2005
, 2005
"... We investigate the application of Semidefinite Programming (SDP) techniques to the VLSI macrocell floorplanning problem. We propose a new mixedinteger SDP formulation of the problem which leads to new SDP relaxations. This approach has been implemented and we report global lower bounds for some MCNC ..."
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Cited by 3 (1 self)
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We investigate the application of Semidefinite Programming (SDP) techniques to the VLSI macrocell floorplanning problem. We propose a new mixedinteger SDP formulation of the problem which leads to new SDP relaxations. This approach has been implemented and we report global lower bounds for some MCNC benchmark macrocell problems. 1