Graph Partitioning into Hamiltonian Subgraphs on a Quantum Annealer

TL;DR

This paper demonstrates the use of a quantum annealer to solve the NP-complete graph partitioning problem into Hamiltonian subgraphs, achieving solutions on large synthetic graphs.

Contribution

It introduces a novel quantum annealing approach for partitioning graphs into Hamiltonian subgraphs and formulates the problem as a quadratic unconstrained binary optimisation.

Findings

Solution probability is independent of cycle length.

Successfully solved graphs with up to 4000 vertices.

Approaches near the physical qubit limit of the quantum annealer.

Abstract

We demonstrate that a quantum annealer can be used to solve the NP-complete problem of graph partitioning into subgraphs containing Hamiltonian cycles of constrained length. We present a method to find a partition of a given directed graph into Hamiltonian subgraphs with three or more vertices, called vertex 3-cycle cover. We formulate the problem as a quadratic unconstrained binary optimisation and run it on a D-Wave Advantage quantum annealer. We test our method on synthetic graphs constructed by adding a number of random edges to a set of disjoint cycles. We show that the probability of solution is independent of the cycle length, and a solution is found for graphs up to 4000 vertices and 5200 edges, close to the number of physical working qubits available on the quantum annealer.