Faster exact solution of sparse MaxCut and QUBO problems

TL;DR

This paper introduces an advanced exact branch-and-cut solver for sparse MaxCut and QUBO problems, significantly outperforming existing methods and solving some instances to optimality for the first time.

Contribution

It develops a parallel, enhanced branch-and-cut algorithm specifically optimized for sparse instances, improving bounds and solving previously unsolved problems.

Findings

Outperforms existing software on sparse MaxCut and QUBO instances

Achieves new best bounds for several benchmark problems

Solves some benchmark instances to optimality for the first time

Abstract

The maximum-cut problem is one of the fundamental problems in combinatorial optimization. With the advent of quantum computers, both the maximum-cut and the equivalent quadratic unconstrained binary optimization problem have experienced much interest in recent years. This article aims to advance the state of the art in the exact solution of both problems -- by using mathematical programming techniques on digital computers. The main focus lies on sparse problem instances, although also dense ones can be solved. We enhance several algorithmic components such as reduction techniques and cutting-plane separation algorithms, and combine them in an exact branch-and-cut solver. Furthermore, we provide a parallel implementation. The new solver is shown to significantly outperform existing state-of-the-art software for sparse MaxCut and QUBO instances. Furthermore, we improve the best known bounds for several instances from the 7th DIMACS Challenge and the QPLIB, and solve some of them (for the first time) to optimality.