紧急顶点覆盖
Emergency Vertex Cover
- Université Paris-Saclay, Univ Evry, IBISC（巴黎萨克雷大学，埃夫里大学，IBISC）
- Université Paris-Saclay, CNRS, LISN（巴黎萨克雷大学，法国国家科学研究中心，LISN）
- LIP6, Sorbonne Université（LIP6，索邦大学）
- Sobolev Institute of Mathematics SB RAS（俄罗斯科学院西伯利亚分院索博列夫数学研究所）
- Research Centre for Operations Research and Statistics, KU Leuven（鲁汶大学运筹学与统计研究中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出紧急顶点覆盖问题，允许远处顶点覆盖边，证明其NP难，给出连续版本多项式算法和离散情况4/3近似算法。
AI中文摘要：
最小顶点覆盖问题是一个基本的组合优化问题，旨在识别图中顶点的最小子集，使得每条边至少与该子集中的一个顶点相关联。在其变体中，最小功率覆盖问题因其实际应用而脱颖而出，例如十字路口的摄像头布置：在边加权图中，如果一条边的某个端点被赋予至少与该边权重一样大的功率值，则该边被覆盖。在本文中，我们引入了紧急顶点覆盖（Em-VC）问题，其中一条边不仅可以通过其端点覆盖，还可以通过一个远处的顶点覆盖，前提是该顶点被赋予足够的功率来“覆盖”沿最短路径到该边一个端点的边的累积权重加上该边的权重。Em-VC受到不同实际场景的启发，例如城市灾害响应的需求，其中确保所有道路段（图的边）的可达性对于有效的援助交付至关重要。我们证明了Em-VC是NP难的，推导出下界，并为其连续版本设计了一个多项式时间算法。此外，我们为离散情况提出了一个4/3近似算法，并确定了几个可以在多项式时间内解决该问题的特殊图类。
英文摘要：
The Minimum Vertex Cover problem is a fundamental combinatorial optimization problem, aiming to identify a minimum subset of vertices in a graph such that every edge is incident to at least one vertex in this subset. Among its variants, the Min-Power-Cover problem stands out due to its practical applications, such as camera placement at intersections: in an edge-weighted graph, an edge is covered if one of its endpoints is assigned a power value at least as large as the edge's weight. In this paper, we introduce the Emergency Vertex Cover (Em-VC) problem where an edge may be covered not only by its endpoints, but also by a distant vertex, provided the vertex is given sufficient power to "cover" the cumulative weight of the edges along a shortest path to one of the edge's endpoints plus the weight of the edge. Em-VC is motivated by different practical scenarios, e.g. the need for urban disaster response, where ensuring accessibility to all road segments (edges of the graph) is crucial for effective aid delivery. We prove that Em-VC is NP-hard, derive lower bounds, and design a polynomial-time algorithm for its continuous version. Moreover, we present a 4/3-approximation algorithm for the discrete case and identify several special graph classes for which the problem can be solved in polynomial time.
