发表机构
Università della Svizzera italiana; Istituto Dalle Molle di studi sull’intelligenza artificiale (IDSIA USI-SUPSI)（瑞士意大利语大学; 达莱莫勒人工智能研究学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出保持积分间隙的归约框架，通过迭代缩小问题实例计算积分间隙，应用于多个优化问题，证明三个简单子类间隙为1并改进其他情况的下界。
AI 中文摘要
我们提出了一个框架，用于系统研究组合优化问题相对于固定线性规划表述的积分间隙。该方法称为“保持积分间隙的归约”，包括迭代缩小问题的输入宇宙，同时保证最大化间隙的实例被保留。当剩余实例的子集变得足够具体时，我们显式计算积分间隙。除了将保持积分间隙的归约应用于三个著名优化问题及其标准线性规划表述（加权顶点覆盖问题、多背包问题和无关机器调度问题）外，我们还通过其配置LP松弛分析了受限分配问题。我们证明了对于该问题的三个“简单”子类，积分间隙等于$1$，这些子类要么在多项式时间内可解，要么承认PTAS（例如，全一处理时间情况）。对于某些剩余情况，我们使用我们的技术改进了当前的下界。
英文摘要
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.
Comments30 pages, 2 figures
