Decomposition algorithms for solving NP-hard problems on a quantum annealer

TL;DR

This paper introduces a decomposition framework for solving NP-hard graph problems like maximum clique and minimum vertex cover on quantum annealers, addressing hardware connectivity limitations through recursive problem division.

Contribution

It presents a novel generic decomposition algorithm with pruning techniques that enables solving NP-hard problems on quantum annealers despite hardware constraints.

Findings

Effective recursive decomposition reduces problem size.

Pruning techniques improve computational efficiency.

Simulation results demonstrate the approach's viability.

Abstract

NP-hard problems such as the maximum clique or minimum vertex cover problems, two of Karp's 21 NP-hard problems, have several applications in computational chemistry, biochemistry and computer network security. Adiabatic quantum annealers can search for the optimum value of such NP-hard optimization problems, given the problem can be embedded on their hardware. However, this is often not possible due to certain limitations of the hardware connectivity structure of the annealer. This paper studies a general framework for a decomposition algorithm for NP-hard graph problems aiming to identify an optimal set of vertices. Our generic algorithm allows us to recursively divide an instance until the generated subproblems can be embedded on the quantum annealer hardware and subsequently solved. The framework is applied to the maximum clique and minimum vertex cover problems, and we propose several pruning and reduction techniques to speed up the recursive decomposition. The performance of both algorithms is assessed in a detailed simulation study.