A Multilevel Bilinear Programming Algorithm For the Vertex Separator Problem

TL;DR

This paper introduces a multilevel bilinear programming algorithm for the Vertex Separator Problem, enhancing previous models by incorporating vertex weights and employing a mountain climbing approach for optimization.

Contribution

It develops a more general continuous bilinear program for the Vertex Separator Problem and integrates a mountain climbing algorithm within a multilevel framework.

Findings

Algorithm outperforms existing methods in computational tests.

Incorporates vertex weights into the bilinear programming model.

Effective in breaking graphs into disconnected components.

Abstract

The Vertex Separator Problem for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets that satisfy specified size constraints. In the paper 10.1016/j.ejor.2014.05.042, the Vertex Separator Problem was formulated as a continuous (non-concave/non-convex) bilinear quadratic program. In this paper, we develop a more general continuous bilinear program which incorporates vertex weights, and which applies to the coarse graphs that are generated in a multilevel compression of the original Vertex Separator Problem. A Mountain Climbing Algorithm is used to find a stationary point of the continuous bilinear quadratic program, while second-order optimality conditions and perturbation techniques are used to escape from either a stationary point or a local maximizer. The algorithms for solving the continuous bilinear program are employed during the solution and refinement phases in a multilevel scheme. Computational results and comparisons demonstrate the advantage of the proposed algorithm.