GAMA: A Novel Algorithm for Non-Convex Integer Programs

TL;DR

GAMA is a classical algorithm leveraging Graver bases for efficiently solving certain non-convex integer programs, outperforming commercial solvers in speed and solution quality on practical problem instances.

Contribution

Introduces GAMA, a novel Graver basis-based algorithm for non-convex integer programs with special structure, demonstrating significant practical efficiency improvements.

Findings

GAMA outperforms Gurobi in solution time by two to three orders of magnitude.

GAMA finds optimal solutions within minutes, unlike Gurobi which takes hours.

The algorithm's slowdown rate with problem size is much lower than traditional solvers.

Abstract

Inspired by the decomposition in the hybrid quantum-classical optimization algorithm we introduced in arXiv:1902.04215, we propose here a new (fully classical) approach to solving certain non-convex integer programs using Graver bases. This method is well suited when (a) the constraint matrix $A$ has a special structure so that its Graver basis can be computed systematically, (b) several feasible solutions can also be constructed easily and (c) the objective function can be viewed as many convex functions quilted together. Classes of problems that satisfy these conditions include Cardinality Boolean Quadratic Problems (CBQP), Quadratic Semi-Assignment Problems (QSAP) and Quadratic Assignment Problems (QAP). Our Graver Augmented Multi-seed Algorithm (GAMA) utilizes augmentation along Graver basis elements (the improvement direction is obtained by comparing objective function values) from these multiple initial feasible solutions. We compare our approach with a best-in-class commercially available solver (Gurobi). Sensitivity analysis indicates that the rate at which GAMA slows down as the problem size increases is much lower than that of Gurobi. We find that for several instances of practical relevance, GAMA not only vastly outperforms in terms of time to find the optimal solution (by two or three orders of magnitude), but also finds optimal solutions within minutes when the commercial solver is not able to do so in 4 or 10 hours (depending on the problem class) in several cases.