Solving the Market Split Problem with Lattice Enumeration

TL;DR

This paper presents a novel lattice enumeration approach to solving the market split problem, significantly extending the solvable instance size compared to traditional methods and recent GPU algorithms.

Contribution

It introduces a lattice enumeration method for the market split problem and demonstrates its effectiveness on benchmark instances up to size 14.

Findings

Lattice enumeration outperforms branch-and-bound on large instances.

The method solves instances up to size 14 on standard hardware.

It surpasses recent GPU-based algorithms in instance size.

Abstract

The market split problem was proposed by Cornu\'ejols and Dawande in 1998 as benchmark problem for algorithms solving linear systems with binary variables. The recent (2025) Quantum Optimization Benchmark Library (QOBLIB) contains a set of feasible instances of the market split problem. The market split problem seems to be difficult to solve with the conventional branch-and-bound approach of integer linear programming software which reportedly can handle QOBLIB instances up to $m=7$. In contrast, a new GPU implementation of the Schroeppel-Shamir algorithm solves instances up to $m=11$. In this short note we report about experiments with our algorithm that reduces the market split problem to a lattice problem. The author's most recent implementation solvediophant applied to the QOBLIB market split instances can solve instances up to $m=14$ on a standard computer.