arXiv:2604.03234cs.AImath.OC2026-04

通过拆解集合覆盖问题的结构,提升启发式算法的求解效率与质量。

Structural Segmentation of the Minimum Set Cover Problem: Exploiting Universe Decomposability for Metaheuristic Optimization

论文配图:Structural Segmentation of the Minimum Set Cover Problem: Exploiting Universe Decomposability for Metaheuristic Optimization
图 1 · 摘自论文原文
  • 利用并查集识别元素共现连通分量,将原问题分解为独立子问题。
  • 在标准测试集上,该方法显著提升解的质量和大规模实例的可扩展性。
  • 适合处理具有明显结构分解特性的大型集合覆盖问题。

最小集合覆盖问题(MSCP)是一类经典的NP难组合优化问题,在科学与工程中有广泛应用。尽管已有大量精确、近似及启发式方法,但多数方法将实例视为整体,忽略了其全域内在结构特性。本文研究了MSCP中的‘全域可分割性’概念,分析如何利用内在结构分解来增强启发式优化。提出一种基于不相交集合并(union-find)的高效预处理策略,通过检测子集中元素的共现连通分量,将原实例分解为独立子问题。每个子问题使用GRASP启发式求解,部分解可直接组合而不破坏可行性。在标准基准实例与大规模合成数据集上的实验表明,利用自然的全域分割能持续提升解的质量与可扩展性,尤其对大型且结构可分解的实例效果显著。该方法采用简洁的位级集合表示,实现高效的集合运算,使方案在大规模场景下具备计算可行性。

原文摘要 · Abstract (English)

The Minimum Set Cover Problem (MSCP) is a classical NP-hard combinatorial optimization problem with numerous applications in science and engineering. Although a wide range of exact, approximate, and metaheuristic approaches have been proposed, most methods implicitly treat MSCP instances as monolithic, overlooking potential intrinsic structural properties of the universe. In this work, we investigate the concept of \emph{universe segmentability} in the MSCP and analyze how intrinsic structural decomposition (universe segmentability) can be exploited to enhance heuristic optimization. We propose an efficient preprocessing strategy based on disjoint-set union (union--find) to detect connected components induced by element co-occurrence within subsets, enabling the decomposition of the original instance into independent subproblems. Each subproblem is solved using the GRASP metaheuristic, and partial solutions are combined without compromising feasibility. Extensive experiments on standard benchmark instances and large-scale synthetic datasets show that exploiting natural universe segmentation consistently improves solution quality and scalability, particularly for large and structurally decomposable instances. These gains are supported by a succinct bit-level set representation that enables efficient set operations, making the proposed approach computationally practical at scale.

组合优化启发式算法结构分解

Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品;中文卡片由大模型生成,请以原文为准。