Faster exact solution of sparse MaxCut and QUBO problems

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. 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 maximum-cut and quadratic unconstrained binary optimization 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.

Efficient Solutions for Large-Scale Max-Cut Problems: A Hybrid Local Search Heuristic Approach and Comparative Analysis with Quantum Annealing

“…Max-Cut problem with many real-world implementations is an NP-hard problem which means that exact solution algorithms are not sufficient for large-scale problems, since obtaining a solution is very time-consuming [28]. Following the advances in quantum computers, max-cut problems have received much interest recently [29]. Two formulations, linear and nonlinear, exist for the MCP [27].…”

Efficient Solutions for Large-Scale Max-Cut Problems: A Hybrid Local Search Heuristic Approach and Comparative Analysis with Quantum Annealing

“…The max-cut problem with many real-world implementations is an NP-hard problem, meaning that exact solution algorithms are insufficient for large-scale problems since obtaining a solution is very time-consuming [28]. Following the advances in quantum computers, max-cut problems have received much interest recently [29]. Two formulations, linear and nonlinear, exist for the MCP [27].…”

Entropic property of randomized QAOA circuits

“…Other two problems are random weighted and unweighted QUBO problems with density (rate of non-zero s ij elements) uniformly distributed in [0.1, 0.9] and integer coefficients uniformly distributed in [−3, 3] (weighted QUBO) or always equal 1 (unweighted QUBO). Note that formally an arbitrary n-bit QUBO problem can be reduced to the Max-Cut with the size n + 1 [26], however, we separate Max-Cut and QUBO because of the different generation of random problems. The adiabatic approach underlying QAOA is also used to solve QUBO problems [27].…”