A Lagrangian relaxation approach to the edge-weighted clique problem

Abstract

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.

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

“…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.…”

“…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].…”

Solving the maximum edge weight clique problem via unconstrained quadratic programming

“…They report computational experience with a set of test problems that have become a standard test bed for other researchers in the area. Hunting et al (2001) report on a Lagrangian Relaxation approach that combines standard Lagrangian methods with cutting planes yielding a new approach to MEWCP. Finally, Sorensen (2004) reports on a new branch and cut method based on new classes of facet defining inequalities for the associated b-clique polytope.…”

Decomposition and Dynamic Cut Generation in Integer Linear Programming

“…Our framework extends this paradigm by considering classes of valid inequalities along with relaxations for which the separation of an arbitrary real vector is difficult but separation of solutions to the relaxation can be accomplished effectively. In addition to the three examples presented here, we have discovered a number of common ILPs with classes of valid inequalities and relaxations that fit into this framework, such as the Generalized Assignment Problem [51], the Edge-Weighted Clique Problem [34], the Traveling Salesman Problem [5], the Knapsack Constrained Circuit Problem [39], the Rectangular Partition Problem [16], the Linear Ordering Problem [15], and the Capacitated Minimum Spanning Tree Problem [24].…”