Evaluating the Performance of Direct Higher-Order Formulations in Combinatorial Optimization Problems

TL;DR

This paper demonstrates that directly solving higher-order combinatorial optimization problems using a PUBO approach yields better solution quality and stability than traditional QUBO methods, without additional computational costs.

Contribution

It provides empirical evidence that directly handling higher-order terms improves solution quality in Ising machine-based optimization, avoiding order reduction complexities.

Findings

PUBO solver outperforms QUBO in solution quality and stability

No significant increase in computational time with PUBO approach

Direct higher-order problem solving avoids order-reduction drawbacks

Abstract

Ising machines, including quantum annealing machines, are promising next-generation computers for combinatorial optimization problems. However, due to hardware limitations, most Ising-type hardware can only solve objective functions expressed in linear or quadratic terms of binary variables. Therefore, problems with higher-order terms require an order-reduction process, which increases the number of variables and constraints and may degrade solution quality. In this study, we evaluate the effectiveness of directly solving such problems without order reduction by using a high-performance simulated annealing-based optimization solver capable of handling polynomial unconstrained binary optimization (PUBO) formulations. We compare its performance against a conventional quadratic unconstrained binary optimization (QUBO) solver on the same hardware platform. As benchmarks, we use the low autocorrelation binary sequence (LABS) problem and the vehicle routing problem with distance balancing, both of which naturally include higher-order interactions. Results show that the PUBO solver consistently achieves superior solution quality and stability compared to its QUBO counterpart, while maintaining comparable computational time and requiring no order-reduction compilation indicating potential advantages of directly handling higher-order terms in practical optimization problems.