Quantum computing and the stable set problem

TL;DR

This paper explores solving the NP-hard stable set problem using D-Wave quantum annealer, introducing post-processing and partitioning techniques to improve solution quality and handle larger graph instances.

Contribution

It formulates the stable set problem as a QUBO for quantum annealing, and develops post-processing and partitioning methods to enhance solution quality and scalability.

Findings

Post-processing significantly improves solution quality.

Partitioning enables handling of larger instances.

Quantum annealer solutions are heuristic and may require refinement.

Abstract

Given an undirected graph, the stable set problem asks to determine the cardinality of the largest subset of pairwise non-adjacent vertices. This value is called the stability number of the graph, and its computation is an NP-hard problem. In this paper, we solve the stable set problem using the D-Wave quantum annealer. By formulating the problem as a quadratic unconstrained binary optimization problem with the penalty method, we show its optimal value equals the graph's stability number for specific penalty values. However, D-Wave's quantum annealer is a heuristic, so the solutions may be far from the optimum and may not represent stable sets. To address these, we introduce a post-processing procedure that identifies samples that could lead to improved solutions. Additionally, we propose a partitioning method to handle larger instances that cannot be embedded on D-Wave's quantum processing unit. Finally, we investigate how different penalty parameter values affect the solutions' quality. Extensive computational results show that the post-processing procedure significantly improves the solution quality, while the partitioning method successfully extends our approach to medium-size instances.