Laplacian eigenvalues and the maximum cut problem

Delorme, Charles; Poljak, Svatopluk

doi:10.1007/bf01585184

Cited by 165 publications

(95 citation statements)

References 13 publications

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“…Alternatively, Q can be taken as the Laplacian matrix of the graph, see, e.g. [6]. The resulting eigenvalue bound and the equivalent semidefinite bound have been recently used in numerical studies and found to do exceptionally well, e.g.…”

Section: (Qap) Max• Q(x) := Trace (Axb -2c)x Tmentioning

confidence: 99%

“…[13]. Moreover, the semidefinite bound has been studied in [6] and shown to have a particular good performance index, see [10].…”

Section: (Qap) Max• Q(x) := Trace (Axb -2c)x Tmentioning

confidence: 99%

“…(See [2] for several examples.) In the case that c = 0, the relaxation to (P) is studied in [6] and shown to have a particular good performance index, see [10]. The relaxation comes from the fact that xt Qx = trace xt Qx = trace Qxx t and, for x E F, Yij = XiXj defines a symmetric, rank one, positive semidefinite matrix Y with diagonal elements 1.…”

mentioning

confidence: 99%

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A recipe for semidefinite relaxation for (0,1)-quadratic programming

1995

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Abstract. We review various relaxations of (0,1)-quadratic programming problems. These include semidefinite programs, parametric trust region problems and concave quadratic maximization. All relaxations that we consider lead to efficiently solvable problems. The main contributions of the paper are the following. Using Lagrangian duality, we prove equivalence of the relaxations in a unified and simple way. Some of these equivalences have been known previously, but our approach leads to short and transparent proofs. Moreover we extend the approach to the case of equality constrained problems by taking the squared linear constraints into the objective function. We show how this technique can be applied to the Quadratic Assignment Problem, the Graph Partition Problem and the Max-Clique Problem. Finally we show our relaxation to be best possible among all quadratic majorants with zero trace.

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Section: (Qap) Max• Q(x) := Trace (Axb -2c)x Tmentioning

confidence: 99%

“…[13]. Moreover, the semidefinite bound has been studied in [6] and shown to have a particular good performance index, see [10].…”

Section: (Qap) Max• Q(x) := Trace (Axb -2c)x Tmentioning

confidence: 99%

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A recipe for semidefinite relaxation for (0,1)-quadratic programming

1995

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“…So far the worst known case is the 5-cycle C 5 , for which the ratio is ϕ(G)/mc(G) = 25+5 √ 131 for all graphs with non-negative weights. The conjecture was confirmed for planar graphs in [35], and other classes of graphs in [36,37]. We recall here the result for planar graphs.…”

Section: Lemma 210mentioning

confidence: 61%

“…Hence the functions given by formulas (13), (14), (26), (35), and (36) are convex (resp. concave), and their minimum (resp.…”

Section: Convexitymentioning

confidence: 99%

Eigenvalues in Combinatorial Optimization

Mohar

Poljak

1993

Combinatorial and Graph-Theoretical Problems in Linear Algebra

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MAX-CUT has a randomized approximation scheme in dense graphs

Véga

1996

Random Struct. Alg.

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A cut in a graph G = (V(G), E(G)) is the boundary 6(S) of some subset S V(G) and the maximum cut problem for G is to find the maximum number of edges in a cut. Let MC(G) denote this maximum. For any given 0 < a < 1, E > 0, and 17, we give a randomized algorithm which runs in a polynomial time and which, when applied to any given graph G on n vertices with minimum degree >an, outputs a cut S(S) of G withWe also show that the proposed method can be used to approximate MAXIMUM ACYCLIC SUBGRAPH in the unweighted case. 0 1996 John Wiley & Sons, Inc.

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Laplacian eigenvalues and the maximum cut problem

Cited by 165 publications

References 13 publications

A recipe for semidefinite relaxation for (0,1)-quadratic programming

A recipe for semidefinite relaxation for (0,1)-quadratic programming

Eigenvalues in Combinatorial Optimization

MAX-CUT has a randomized approximation scheme in dense graphs

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