Path Relinking Scheme for the Max-Cut Problem

Path Relinking Scheme for the Max-Cut Problem

Volodymyr P. Shylo (V. M. Glushkov Institute of Cybernetics (IC), Ukraine) and Oleg V. Shylo (University of Tennessee, USA)
Copyright: © 2015 |Pages: 11
DOI: 10.4018/978-1-4666-6328-2.ch011

Abstract

In this chapter, a path relinking method for the maximum cut problem is investigated. The authors consider an implementation of the path-relinking, where it is utilized as a subroutine for another meta-heuristic search procedure. Particularly, the authors focus on the global equilibrium search method to provide a set of high quality solutions, the set that is used within the path relinking method. The computational experiment on a set of standard benchmark problems is provided to study the proposed approach. The authors show that when the size of the solution set that is passed to the path relinking procedure is too large, the resulting running times follow the restart distribution, which guarantees that an underlying algorithm can be accelerated by removing all of the accumulated data (set P) and re-initiating its execution after a certain number of elite solutions is obtained.
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Background

Path relinking method searches for solutions of an optimization problem along the trajectories that connect solutions from a given set (Glover, Laguna & Marti, 2000). The most common path relinking scheme involves a pair of solutions: an initiating solution and a guiding solution. A set of moves (transformations) are applied starting in the initiating solution that sequentially introduce the attributes of the guiding solution. Usually, such moves result in a sequence of solutions that lie on a path between the initial solution pair.

Assuming that the weights of the graph edges are non-negative, the maximum cut problem can be formulated by the following mixed-integer program (Kahruman, Kolotoglu, Butenko & Hicks, 2007):

The optimal solution vector x defines a graph partition (if then , otherwise ) that has the maximum cut value, the sum of weight of edges connecting different partititions. Let denote a cost of a cut corresponding to the solution vector x.

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