Emergency Vertex Cover

2026-09-17

AI Summary

本文提出了紧急顶点覆盖问题(Em-VC)，通过给定足够功率使远端顶点也能覆盖边，旨在解决如城市灾难响应等实际问题，并为此设计了多项式时间算法及近似算法。

Abstract

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.

Problem

• Emergency Vertex Cover
• Min-Power-Cover
• edge-weighted graph

Innovation

• Emergency Vertex Cover
• NP-hard
• polynomial-time algorithm
• 4/3-approximation algorithm