An Analysis of Quantum Annealing Algorithms for Solving the Maximum Clique Problem
Alessandro Gherardi, Alberto Leporati
TL;DR
This work analyzes how quantum annealing on a D-Wave device can solve the maximum clique problem by framing it as a QUBO/Ising problem under a practical embedding limit of $164$ qubits. It introduces an IS-decomposition algorithm to reduce problem size without losing the maximum independent set and a controllable graph-generation method to produce benchmark graphs, enabling a rigorous statistical analysis of how graph structure impacts solution quality. The experiments identify a critical threshold near $0.42$ for the ratio of clique size to external nodes, beyond which solution quality degrades, and report empirical laws linking clique size to proximity to optimum, while showing that density and standard connectivity indices offer limited predictive power. Practically, the results guide instance design and preprocessing for current hardware, suggesting near-$100$-node graphs yield more reliable results before embedding limits are reached, and motivate hardware scaling to handle larger, harder instances.
Abstract
Quantum annealers can be used to solve many (possibly NP-hard) combinatorial optimization problems, by formulating them as quadratic unconstrained binary optimization (QUBO) problems or, equivalently, using the Ising formulation. In this paper we analyse the ability of quantum D-Wave annealers to find the maximum clique on a graph, expressed as a QUBO problem. Due to the embedding limit of 164 nodes imposed by the anneler, we conducted a study on graph decomposition to enable instance embedding. We thus propose a decomposition algorithm for the complementary maximum independent set problem, and a graph generation algorithm to control the number of nodes, the number of cliques, the density, the connectivity indices and the ratio of the solution size to the number of other nodes. We then statistically analysed how these variables affect the quality of the solutions found by the quantum annealer. The results of our investigation include recommendations on ratio and density limits not to be exceeded, as well as a series of precautions and a priori analyses to be carried out in order to maximise the probability of obtaining a solution close to the optimum.