On bounding the bandwidth of graphs with symmetry

TL;DR

This paper introduces new bounds for graph bandwidth using a strengthened semidefinite programming relaxation of the minimum cut problem, exploiting symmetry to improve bounds on various symmetric graphs.

Contribution

It presents a novel SDP relaxation for the minimum cut problem that leverages symmetry, providing improved bounds for graph bandwidth, and develops a heuristic based on reverse Cuthill-McKee.

Findings

Achieved the best known bounds for Hamming, Johnson, and Kneser graphs.

The SDP relaxation effectively exploits symmetry to tighten bounds.

The heuristic significantly improves upper bounds on tested graphs.

Abstract

We derive a new lower bound for the bandwidth of a graph that is based on a new lower bound for the minimum cut problem. Our new semidefinite programming relaxation of the minimum cut problem is obtained by strengthening the known semidefinite programming relaxation for the quadratic assignment problem (or for the graph partition problem) by fixing two vertices in the graph; one on each side of the cut. This fixing results in several smaller subproblems that need to be solved to obtain the new bound. In order to efficiently solve these subproblems we exploit symmetry in the data; that is, both symmetry in the min-cut problem and symmetry in the graphs. To obtain upper bounds for the bandwidth of graphs with symmetry, we develop a heuristic approach based on the well-known reverse Cuthill-McKee algorithm, and that improves significantly its performance on the tested graphs. Our approaches result in the best known lower and upper bounds for the bandwidth of all graphs under consideration, i.e., Hamming graphs, 3-dimensional generalized Hamming graphs, Johnson graphs, and Kneser graphs, with up to 216 vertices.