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

TL;DR

This work evaluates direct polynomial unconstrained binary optimization (PUBO) against the standard quadratic (QUBO) approach on Ising-machine hardware, using the Fixstars Amplify Annealing Engine to solve problems with higher-order interactions. Across two benchmarks—low autocorrelation binary sequences (LABS) and a distance-balanced vehicle routing problem (VRP)—PUBO demonstrates superior solution quality and stability, with runtimes comparable to QUBO and no order-reduction overhead. The study also analyzes how order-reduction penalties inflate problem size and degrade performance, showing PUBO preserves problem structure and scales better as problem size grows. The results suggest direct PUBO formulations can offer practical advantages in real-world higher-order optimization and motivate hardware and solver developments to support PUBO directly.

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.

Figures (8)

• Figure 1: Conceptual illustration of the correlation structure in the LABS problem for $N=5$. Each node represents a spin variable $s_i \in \{-1, +1\}$. Edges of the same color correspond to spin pairs contributing to the autocorrelation term $C_k$ at lag $k$, where red, blue, green, and purple represent shifts of $k = 1, 2, 3,$ and $4$, respectively. The overall energy $E = \sum_{k=1}^{N-1} C_k^2$ is given by the sum of squared correlations at different lags, indicating that the interactions among spins arise from the squared combination of these color-coded pairwise correlations.
• Figure 2: Conceptual illustration of the vehicle routing problem. In this example, one central depot is connected to ten distinct customer locations. The task is to design delivery routes such that all customer locations are visited, starting and ending at the depot. Here, an example using four vehicles to complete the deliveries is shown.
• Figure 3: Relationship between the sequence length $N$ and average normalized energy $\tilde{E}$ in the LABS problems. The plot shows the mean of ten runs, and the shaded area represents the standard deviation.
• Figure 4: Results for a single VRP instance with $P=11$ customer locations and $V=3$ vehicles. Shown are the relationships between the hyperparameter $\alpha$ and (a) total travel distance, (b) variance of travel distances, (c) objective function value. Each curve represents the mean of ten independent runs with a runtime of 60 seconds, and the shaded areas represent standard deviations.
• Figure 5: Scatter plot showing the trade-off between the two objective values. The x-axis represents the total travel distance and the y-axis represents the variance of travel distances among vehicles. The results are obtained from 10 independent runs for each of 100 different values of $\alpha$. Only feasible solutions that satisfy all constraints are shown, while infeasible ones are omitted.