Quantum Speedups for Group Relaxations of Integer Linear Programs
🤖 AI Summary
Integer linear programming (ILP) is an NP-hard problem, and existing quantum algorithms struggle to achieve super-quadratic speedups in settings with multiple constraints. This work proposes a novel approach that integrates Gomory group relaxation with constraint-preserving local search, constructing—for the first time—a constraint-preserving mixer for ILP group relaxations with favorable spectral properties, thereby enabling super-quadratic quantum acceleration. Under non-degeneracy conditions, the method exactly recovers the optimal solution to the original problem; otherwise, it effectively narrows the integrality gap, substantially enhancing the performance of branch-and-cut algorithms. Empirical evaluations demonstrate the efficacy of the proposed method across multiple real-world ILP instances.
Application Category
📝 Abstract
Integer Linear Programs (ILPs) are a flexible and ubiquitous model for discrete optimization problems. Solving ILPs is \textsf{NP-Hard} yet of great practical importance. Super-quadratic quantum speedups for ILPs have been difficult to obtain because classical algorithms for many-constraint ILPs are global and exhaustive, whereas quantum frameworks that offer super-quadratic speedup exploit local structure of the objective and feasible set. We address this via quantum algorithms for Gomory's group relaxation. The group relaxation of an ILP is obtained by dropping nonnegativity on variables that are positive in the optimal solution of the linear programming (LP) relaxation, while retaining integrality of the decision variables. We present a competitive feasibility-preserving classical local-search algorithm for the group relaxation, and a corresponding quantum algorithm that, under reasonable technical conditions, achieves a super-quadratic speedup. When the group relaxation satisfies a nondegeneracy condition analogous to, but stronger than, LP non-degeneracy, our approach yields the optimal solution to the original ILP. Otherwise, the group relaxation tightens bounds on the optimal objective value of the ILP, and can improve downstream branch-and-cut by reducing the integrality gap; we numerically observe this on several practically relevant ILPs. To achieve these results, we derive efficiently constructible constraint-preserving mixers for the group relaxation with favorable spectral properties, which are of independent interest.
Problem
Research questions and friction points this paper is trying to address.
Integer Linear Programs
Quantum Speedup
Group Relaxation
NP-Hard Optimization
Integrality Gap
Innovation
Methods, ideas, or system contributions that make the work stand out.
quantum speedup
group relaxation
integer linear programming
constraint-preserving mixer
super-quadratic acceleration
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