Solving Large Maximum Clique Problems on a Quantum Annealer

Abstract

Commercial quantum annealers from D-Wave Systems can find high quality solutions of quadratic unconstrained binary optimization problems that can be embedded onto its hardware. However, even though such devices currently offer up to 2048 qubits, due to limitations on the connectivity of those qubits, the size of problems that can typically be solved is rather small (around 65 variables). This limitation poses a problem for using D-Wave machines to solve application-relevant problems, which can have thousands of variables. For the important Maximum Clique problem, this article investigates methods for decomposing larger problem instances into smaller ones, which can subsequently be solved on D-Wave. During the decomposition, we aim to prune as many generated subproblems that do not contribute to the solution as possible, in order to reduce the computational complexity. The reduction methods presented in this article include upper and lower bound heuristics in conjunction with graph decomposition, vertex and edge extraction, and persistency analysis.

Using Machine Learning for Quantum Annealing Accuracy Prediction

“…During training, we noticed that the decision tree machine learning model appears to have trouble with problems that return a clique size of zero. This was due to a suboptimal QUBO formulation we used for MC, and the problem disappeared when using the formulation of [1,4]. However, similar issues might occur with other problems, highlighting that the present approach might be most suitable for problem classes that can be solved sufficiently well on D-Wave.…”

“…The task of finding a maximum clique has to be mapped to a formulation of the type of Equation (1). As shown in [1,4], the QUBO function…”

“…x i ∈ {0, 1} for all i ∈ V, achieves a minimum if and only if the vertices i ∈ V for which x i = 1 form a maximum clique, where the two constants can be chosen as A = 1, B = 2 [4]. It is noteworthy that any function of the type of Equation ( 1) can be represented as a graph P itself, which is used to embed the QUBO problem into the quantum processor.…”

Solving large minimum vertex cover problems on a quantum annealer

“…A decomposition algorithm for another NP-hard graph problem, the maximum clique problem, has been proposed in Chapuis et al (2017); Pelofske et al (2019). In Pelofske et al (2019), the authors additionally investigate a variety of techniques to prune subproblems during the recursive decomposition, for instance by computing bounds on the clique size.…”

“…A decomposition algorithm for another NP-hard graph problem, the maximum clique problem, has been proposed in Chapuis et al (2017); Pelofske et al (2019). In Pelofske et al (2019), the authors additionally investigate a variety of techniques to prune subproblems during the recursive decomposition, for instance by computing bounds on the clique size. Similarly, to solve the maximum independent set problem, an equivalent formulation of the maximum clique problem, several algorithms are known including some relying on graph decomposition (Giakoumakis and Vanherpe, 1997;Courcelle et al, 2000).…”