An Analysis of Quantum Annealing Algorithms for Solving the Maximum Clique Problem

TL;DR

This paper evaluates the effectiveness of quantum annealers in solving the maximum clique problem, focusing on graph decomposition techniques and problem instance characteristics to optimize solution quality.

Contribution

It introduces a novel graph decomposition algorithm and provides guidelines for problem instance parameters to improve quantum annealing performance.

Findings

Quantum annealers can find near-optimal solutions within certain graph density and ratio limits.

Decomposition techniques enable embedding larger graphs into quantum annealers.

Recommendations are provided for problem instance design to maximize solution quality.

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.