An extended formulation approach to the edge-weighted maximal clique problem

Park, Kyungchul; Lee, Kyungsik

doi:10.1016/0377-2217(95)00299-5

Cited by 55 publications

(43 citation statements)

References 12 publications

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“…Nevertheless, interesting special cases enjoying this property exist. Examples are the "trivial" inequalities mentioned in the following result due to Padberg [1989] and Park et al [1996]. The following proposition exhibits two more classes of inequalities that are facet-defining for both QP n and CP n b .…”

Section: 1mentioning

confidence: 84%

“…Hence equality holds. Substitution of this equality into (25) with α = p now yields the desired result.…”

Section: Lemma 41 a (0 1)-vector (X Y) Satisfies The Constraints mentioning

confidence: 93%

“…Exact branch-and-cut algorithms for (W CP b ), based on linear programming have been proposed by Dijkhuizen and Faigle [1993], Park et al [1996], Mehrotra [1997], and Macambira and De Souza [1997]. In these four papers several facet-defining inequalities are introduced which are used as cutting planes in order to improve the quality of the upper bounds.…”

Section: Introductionmentioning

confidence: 99%

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A Lagrangian relaxation approach to the edge-weighted clique problem

Hunting

Faigle

Kern

2001

European Journal of Operational Research

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Abstract. The b-clique polytope CP n b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QP n as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over QP n n . The problem of optimizing linear functions over CP n b has so far been approached via heuristic combinatorial algorithms and cutting-plane methods.We study the structure of CP n b in further detail and present a new computational approach to the linear optimization problem based on Lucena's suggestion of integrating cutting planes into a Lagrangian relaxation of an integer programming problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented.

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Section: 1mentioning

confidence: 84%

“…Hence equality holds. Substitution of this equality into (25) with α = p now yields the desired result.…”

Section: Lemma 41 a (0 1)-vector (X Y) Satisfies The Constraints mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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A Lagrangian relaxation approach to the edge-weighted clique problem

Hunting

Faigle

Kern

2001

European Journal of Operational Research

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“…We refer the reader to [19,7,13,11,9,8,14,3,1] for more details on other existing facet defining inequalities and solution methods for the MEWCP. For the sake of brevity, we restrict our literature review to the paper of Djeumou Fomeni et al [4] in which they presented the cutting planes that are discussed in this paper.…”

Section: Literature Reviewmentioning

confidence: 99%

“…Ideally, one would like to use cutting planes that are facet defining and computationally 'easy' to generate. Several families of facet defining inequalities are proposed in the literature for this purpose, see for example [19,7,13,11,9,8,14].…”

Section: Introductionmentioning

confidence: 99%

A new family of facet defining inequalities for the maximum edge-weighted clique problem

Fomeni

2016

Optim Lett

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This paper considers a family of cutting planes, recently developed for mixed 0-1 polynomial programs and shows that they define facets for the maximum edge-weighted clique problem. There exists a polynomial time exact separation algorithm for these inequalities. The result of this paper may contribute to the development of more efficient algorithms for the maximum edge-weighted clique problem that use cutting planes.

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Concise RLT forms of binary programs: A computational study of the quadratic knapsack problem

Forrester

Adams

Hadavas

2009

Naval Research Logistics

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Abstract:The reformulation-linearization technique (RLT) is a methodology for constructing tight linear programming relaxations of mixed discrete problems. A key construct is the multiplication of "product factors" of the discrete variables with problem constraints to form polynomial restrictions, which are subsequently linearized. For special problem forms, the structure of these linearized constraints tends to suggest that certain classes may be more beneficial than others. We examine the usefulness of subsets of constraints for a family of 0-1 quadratic multidimensional knapsack programs and perform extensive computational tests on a classical special case known as the 0-1 quadratic knapsack problem. We consider RLT forms both with and without these inequalities, and their comparisons with linearizations derived from published methods. Interestingly, the computational results depend in part upon the commercial software used.

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An extended formulation approach to the edge-weighted maximal clique problem

Cited by 55 publications

References 12 publications

A Lagrangian relaxation approach to the edge-weighted clique problem

A Lagrangian relaxation approach to the edge-weighted clique problem

A new family of facet defining inequalities for the maximum edge-weighted clique problem

Concise RLT forms of binary programs: A computational study of the quadratic knapsack problem

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