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 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.

Improved local models and new Bell inequalities via Frank-Wolfe algorithms

“…(B1) is NP-hard and, to the best of our knowledge, the largest general instance (that is, not exploiting a very specific form of M as in [9], for instance) solved in the literature is in [13], and involves a matrix M of size 90 × 90, and uses a custom branch-and-bound algorithm. Here we argue that the recent developments to solve QUBO problems make the result therein obsolete, as the problem can be reformulated to exploit general solvers [35].…”

“…The difficulty to improve on these works lies in the computation of the local value of the Bell inequality provided by the algorithm [32][33][34]. Interestingly, however, this problem can be converted into a Quadratic Unconstrained Binary Optimisation (QUBO) instance, a class of problems which has seen some recent improvements, see [35] and references therein.…”

“…We had access to a version of the solver from [35]. With an initial set of 97 measurements, we ran our algorithm starting from v 0 = 0.6958 and fed the QUBO solver with the resulting Bell inequality in order to obtain, in about half an hour, the bound…”

“…We refer to Appendix B for details, including the reformulation of the computation as a QUBO. Importantly, this bound is also analytical as the Bell inequality used has integer entries, so that the decision made in [35] to cut branches are exact.…”

“…Acknowledgements.-The authors are grateful to Flavien Hirsch for his assistance in drawing the connection between Frank-Wolfe algorithms and Gilbert's algorithm, to Daniel Rehfeldt for allowing us to utilize his QUBO solver [35], to Tamás Vértesi for supplying the data we used to verify our algorithm, and to Nicolas Brunner, Máté Farkas, Antonio Frangioni, and Leo Liberti for their contribution in the discussions. This research was partially funded by the DFG Cluster of Excellence MATH+ (EXC-2046/1, project id 390685689) funded by the Deutsche Forschungsgemeinschaft (DFG).…”