A Multilevel Bilinear Programming Algorithm For the Vertex Separator Problem

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. The Vertex Separator Problem was formulated in the paper 10.1016/j.ejor.2014.05.042 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 amultilevel compression of the originalVertex Separator Problem.We develop amethod for improving upon a given vertex separator by applying aMountain Climbing Algorithm to the bilinear program using an incidence vector for the separator as a starting guess. Sufficient conditions are developed under which the algorithm September 1st, 2017. The research was supported by the Office of Naval Research under Grants N00014-11-1-0068 and N00014-15-1-2048 and by the National Science Foundation under Grants 1522629 and 1522751. Part of the research was performed while the second author was a Givens Associate at Argonne National Laboratory. B James T. Hungerford [email protected] William W. Hager [email protected] http://people.clas.ufl.edu/hager/ Ilya Safro [email protected] http://www.cs.clemson.edu/∼isafro 1 Department of Mathematics, University of Florida, PO Box 118105, Gainesville, FL 32611-8105, USA 2 RaceTrac Store Support Center, 200 Galleria Pkwy SE, Atlanta, GA 30339, USA 3 School of Computing, Clemson University, 228 McAdams Hall, Clemson, SC 29634, USA