Quadratic Convex Reformulation : a Computational Study of the Graph Bisection Problem
Alain Billionnet, Sourour Elloumi, Marie-Christine Plateau · 2005
Given an undirected graph G = (V, E), we consider the graph bisection problem, which consists in partitioning the nodes of G in two disjoined sets with p and n − p nodes respectively such that the total weight of edges crossing between subsets is minimal. We apply QCR to it, a general method, presented in [4], which combines semidefinite programming (SDP) and Mixed Integer Quadratic Programming (MIQP). This method solves exactly general 0-1 quadratic programs (called (QP)) with linear constraints. It is composed of two phases: Phase I is devoted to reformulate (QP) into a new problem, equivalent to (QP), with a convex quadratic function; Phase II consists in submitting this new problem to a mixed-integer convex quadratic solver. Computational results reported in this paper show that the present approach is competitive with state-of-the-art methods for the graph bisection problem.