Integrality gap preserving reductions

2026-09-17

AI Summary

研究通过整数间隙保持约简方法解决组合优化问题的整数间隙，应用于多个经典问题，并证明某些子类问题的整数间隙为1。

Abstract

We propose a framework for the systematic study of integrality gaps of combinatorial optimization problems with respect to a fixed linear programming formulation. The method, called \emph{integrality gap preserving reduction}, consists of iteratively shrinking the input universe of the problem while guaranteeing that gap-maximizing instances remain selected. When the subset of remaining instances becomes specific enough, we calculate the integrality gap explicitly. Besides applying integrality gap preserving reductions to three well-known optimization problems via their standard linear programming formulations (weighted vertex cover problem, multiple knapsack problem, and unrelated machine scheduling problem), we analyse the restricted assignment problem via its configuration LP relaxation. We prove that the integrality gap is equal to $1$ for three ``easy'' subclasses of the problem that are either solvable in polynomial time or admit a PTAS (e.g., the all-one processing time case). For some remaining cases, we improve the current lower bound using our technique.

Problem

• combinatorial optimization
• integrality gap
• linear programming
• reduction

Innovation

• integrality gap preserving reduction
• combinatorial optimization
• linear programming formulation
• configuration LP relaxation
• restricted assignment problem