Many industrially relevant problems can be deterministically solved by computers in principle, but are intractable in practice, as the seminal P/NP dichotomy of complexity theory and Cobham’s thesis testify. For the many NP-complete problems, industry needs to resort to using heuristics or approximation algorithms. For approximation algorithms, there is a more refined classification in complexity classes that goes beyond the simple P/NP dichotomy. As it is well known, approximation classes form a hierarchy, that is, FPTAS  PTAS  APX  NPO. This classification gives a more realistic notion of complexity but—unless unexpected breakthroughs happen for fundamental problems like P = NP or related questions— there is no known efficient algorithm that can solve such problems exactly on a realistic computer. Therefore, new ways of computations are sought. Recently, considerable hope was placed on the possible computational powers of quantum computers and quantum annealing (QA) in particular. However, the precise benefits of such a drastic shift in hardware are still unchartered territory to a good extent. Firstly, the exact relations between classical and quantum complexity classes pose many open questions, and secondly, technical details of formulating and implementing quantum algorithms play a crucial role in real-world applications. Guided by the hierarchy of classical optimisation complexity classes, we discuss how to map problems of each class to a quantum annealer. Those problems are the Minimum Multiprocessor Scheduling (MMS) problem, the Minimum Vertex Cover (MVC) problem and the Maximum Independent Set (MIS) problem. We experimentally investigate if and how the degree of approximability influences implementation and run-time performance. Our experiments indicate a discrepancy between classical approximation complexity and QA behaviour: Problems MIS and MVC, members of APX respectively PTAS, exhibit better solution quality on a QA than MMS, which is in FPTAS, even despite the use of preprocessing the for latter. This leads to the hypothesis that traditional classifications do not immediately extend to the quantum annealing domain, at least when the properties of real-world devices are taken into account. A structural reason, why FPTAS problems do not show good solution quality, might be the use of an inequlity in the problem description of the FPTAS problems. Formulating those inequalities on a quantum hardware (mostly done by formulating a Quadratic Unconstrained Binary optimisation (QUBO) problem in form of a matrix) requires a lot of hardware space which makes finding an optimal solution more difficult. Reducing the density of a QUBO is possible by appropriately pruning QUBO matrices. For the problems considered in our evaluation, we find that the achievable solution quality on a real-world machine is unexpectedly robust against pruning, often up to ratios as high as 50% or more. Since quantum annealers are probabilistic machines by design, the loss in solution quality is only of subordinate relevance, especially considering that the pruning of QUBO matrices allows for solving larger problem instances on hardware of a given capacity. We quantitatively discuss the interplay between these factors.

1st International Symposium on Applied Artificial Intelligence (ISAAI’19)

The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem (NPO) that has been of great interest for decades for both, science and industry. The CVRP is a variant of the vehicle routing problem characterized by capacity constrained vehicles. The aim is to plan tours for vehicles to supply a given number of customers as efficiently as possible. The problem is the combinatorial explosion of possible solutions, which increases superexponentially with the number of customers. Classical solutions provide good approximations to the globally optimal solution. D-Wave’s quantum annealer is a machine designed to solve optimization problems. This machine uses quantum effects to speed up computation time compared to classic computers. The problem on solving the CVRP on the quantum annealer is the particular formulation of the optimization problem. For this, it has to be mapped onto a quadratic unconstrained binary optimization (QUBO) problem. Complex optimization problems such as the CVRP can be translated to smaller subproblems and thus enable a sequential solution of the partitioned problem. This work presents a quantum-classic hybrid solution method for the CVRP. It clarifies whether the implementation of such a method pays off in comparison to existing classical solution methods regarding computation time and solution quality. Several approaches to solving the CVRP are elaborated, the arising problems are discussed, and the results are evaluated in terms of solution quality and computation time.

Frontiers in ICT, 2019, pp. 1-13