Cutting Slack: Quantum Optimization with Slack-Free Methods for Combinatorial Benchmarks

TL;DR

This paper explores slack-free Lagrangian methods for quantum combinatorial optimization, reducing qubit requirements and improving scalability for NP-hard problems like TSP, MDKP, and MIS.

Contribution

It introduces a suite of Lagrangian-based techniques that eliminate the need for slack variables, enabling more efficient quantum optimization of constrained problems.

Findings

Slack-free reformulations reduce qubit counts for TSP and MDKP.

Lagrangian methods improve feasibility and solution quality.

Trade-offs between qubit savings and optimality are analyzed.

Abstract

Constraint handling remains a key bottleneck in quantum combinatorial optimization. While slack-variable-based encodings are straightforward, they significantly increase qubit counts and circuit depth, challenging the scalability of quantum solvers. In this work, we investigate a suite of Lagrangian-based optimization techniques including dual ascent, bundle methods, cutting plane approaches, and augmented Lagrangian formulations for solving constrained combinatorial problems on quantum simulators and hardware. Our framework is applied to three representative NP-hard problems: the Travelling Salesman Problem (TSP), the Multi-Dimensional Knapsack Problem (MDKP), and the Maximum Independent Set (MIS). We demonstrate that MDKP and TSP, with their inequality-based or degree-constrained structures, allow for slack-free reformulations, leading to significant qubit savings without compromising performance. In contrast, MIS does not inherently benefit from slack elimination but still gains in feasibility and objective quality from principled Lagrangian updates. We benchmark these methods across classically hard instances, analyzing trade-offs in qubit usage, feasibility, and optimality gaps. Our results highlight the flexibility of Lagrangian formulations as a scalable alternative to naive QUBO penalization, even when qubit savings are not always achievable. This work provides practical insights for deploying constraint-aware quantum optimization pipelines, with applications in logistics, network design, and resource allocation.