Connectivity via convexity: Bounds on the edge expansion in graphs

TL;DR

This paper introduces a convexification-based approach using a doubly non-negative relaxation and an augmented Lagrangian algorithm to compute bounds on the edge expansion of graphs, a challenging combinatorial problem.

Contribution

It applies a novel convexification technique to formulate and relax the edge expansion problem as a completely positive program, providing efficient bounds.

Findings

Relaxation yields strong bounds on edge expansion.

Method is computationally efficient for graphs with hundreds of vertices.

Augmented Lagrangian algorithm effectively solves the relaxation.

Abstract

Convexification techniques have gained increasing interest over the past decades. In this work, we apply a recently developed convexification technique for fractional programs by He, Liu and Tawarmalani (2024) to the problem of determining the edge expansion of a graph. Computing the edge expansion of a graph is a well-known, difficult combinatorial problem that seeks to partition the graph into two sets such that a fractional objective function is minimized. We give a formulation of the edge expansion as a completely positive program and propose a relaxation as a doubly non-negative program, further strengthened by cutting planes. Additionally, we develop an augmented Lagrangian algorithm to solve the doubly non-negative program, obtaining lower bounds on the edge expansion. Numerical results confirm that this relaxation yields strong bounds and is computationally efficient, even for graphs with several hundred vertices.