Adaptive Graph Shrinking for Quantum Optimization of Constrained Combinatorial Problems

TL;DR

This paper introduces a hybrid classical-quantum graph shrinking method to improve quantum optimization of constrained combinatorial problems, addressing hardware limitations and enhancing solution feasibility.

Contribution

It presents a novel adaptive graph shrinking framework with constraint-awareness and verification, tailored for quantum optimization of complex combinatorial problems.

Findings

Improves solution feasibility and reduces repair complexity.

Enhances quantum optimization quality on hardware-limited instances.

Demonstrates scalability for near-term quantum algorithms.

Abstract

A range of quantum algorithms, especially those leveraging variational parameterization and circuit-based optimization, are being studied as alternatives for solving classically intractable combinatorial optimization problems (COPs). However, their applicability is limited by hardware constraints, including shallow circuit depth, limited qubit counts, and noise. To mitigate these issues, we propose a hybrid classical--quantum framework based on graph shrinking to reduce the number of variables and constraints in QUBO formulations of COPs, while preserving problem structure. Our approach introduces three key ideas: (i) constraint-aware shrinking that prevents merges that will likely violate problem-specific feasibility constraints, (ii) a verification-and-repair pipeline to correct infeasible solutions post-optimization, and (iii) adaptive strategies for recalculating correlations and controlling the graph shrinking process. We apply our approach to three standard benchmark problems: Multidimensional Knapsack (MDKP), Maximum Independent Set (MIS), and the Quadratic Assignment Problem (QAP). Empirical results show that our approach improves solution feasibility, reduces repair complexity, and enhances quantum optimization quality on hardware-limited instances. These findings demonstrate a scalable pathway for applying near-term quantum algorithms to classically challenging constrained optimization problems.