Quantum computing and the stable set problem

TL;DR

This work tackles the NP-hard stable set problem by formulating it as a QUBO with a penalty parameter $\beta$, proving that for exact solvers the optimum equals the graph's stability number $\alpha(G)$ whenever $\beta\ge \tfrac{1}{2}$. Given the heuristic nature of D-Wave quantum annealing, the authors introduce a post-processing pipeline and a CH-partitioning-based strategy to obtain high-quality feasible stable sets on larger graphs, complemented by an annihilation-number bound to prune subproblems. Extensive experiments on DIMACS complements, Paley, and evil instances demonstrate that post-processing substantially improves solution quality, with $\beta=\tfrac{1}{2}$ often providing the best balance between solution quality and computational effort. The combination of QUBO formulation, post-processing, and simple partitioning enables solving medium-to-large instances on quantum hardware or hybrids, offering a scalable route toward practical quantum-accelerated stable-set computations and insights into parameter choices and problem decomposition.

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.

Figures (1)

• Figure 1: The left figure shows a graph $G$ on five vertices. In this graph, the following subsets of vertices are stable sets: $\{v_1, v_2\}$, $\{v_1, v_3\}$, $\{v_1, v_5\}$, $\{v_1, v_3, v_5\}$, $\{v_3, v_5\}$, $\{v_4, v_5\}$. The right figure shows the complement graph $\overline{G}$, where these same subsets form cliques. In particular, the maximum stable set $\{v_1, v_3, v_5\}$ in $G$ is equivalent to the maximum clique in $\overline{G}$, and therefore, $\alpha(G) = \omega(\overline{G}) = 3$.

Theorems & Definitions (20)

• Lemma 2
• proof
• Example 3
• Corollary 4
• proof
• Proposition 5
• proof
• Example 6
• Lemma 7
• proof