Extending Sample Persistence Variable Reduction for Constrained Combinatorial Optimization Problems

TL;DR

Constrained COPs map to QUBO with energy $\mathcal{H}(\bm{x}) = \mathcal{H}_{\mathrm{obj}}(\bm{x}) + \sum_m \mu_m \mathcal{H}_{\mathrm{const}}^{(m)}(\bm{x})$, balancing feasibility and objective value. To address hardware limits, the paper extends SPVAR to MP-SPVAR, which aggregates solution persistence across multiple penalty coefficients to fix variables. Experiments on QAP and QKP show MP-SPVAR achieves higher feasible-solution ratios and maintains or improves approximation ratios compared with SPVAR. Analysis of low-energy states under small penalties clarifies when feasibility degrades and how encoding choices affect the trade-off, positioning MP-SPVAR as a practical reduction strategy and a basis for penalty-tuning and hardware integration.

Abstract

Constrained combinatorial optimization problems (CCOPs) are challenging to solve due to the exponential growth of the solution space. When tackled with Ising machines, constraints are typically enforced by the penalty function method, whose coefficients must be carefully tuned to balance feasibility and objective quality. Variable-reduction techniques such as sample persistence variable reduction (SPVAR) can mitigate hardware limitations of Ising machines, yet their behavior on CCOPs remains insufficiently understood. Building on our prior proposal, we extend and comprehensively evaluate multi-penalty SPVAR (MP-SPVAR), which fixes variables using solution persistence aggregated across multiple penalty coefficients. Experiments on benchmark problems, including the quadratic assignment problem and the quadratic knapsack problem, demonstrate that MP-SPVAR attains higher feasible-solution ratios while matching or improving approximation ratios relative to the conventional SPVAR algorithm. An examination of low-energy states under small penalties clarifies when feasibility degrades and how encoding choices affect the trade-off between solution quality and feasibility. These results position MP-SPVAR as a practical variable-reduction strategy for CCOPs and lay a foundation for systematic penalty tuning, broader problem classes, and integration with quantum-inspired optimization hardware as well as quantum algorithms.

Figures (46)

• Figure 1: Illustration of the MP-SPVAR algorithm. The black, white circles represent the variable that takes the value 1 and 0, respectively, and the gray circle represents the undetermined variable.
• Figure 2:
• Figure 3:
• Figure 5:
• Figure 6: