On Colorful Vertex and Edge Cover Problems
Sayan Bandyapadhyay, Aritra Banik, Sujoy Bhore

TL;DR
This paper introduces colorful variants of vertex and edge cover problems, providing approximation algorithms for the vertex version and exact algorithms for the edge version, with applications in fair geometric set systems.
Contribution
It presents a $(2+ ext{epsilon})$-approximation for colorful vertex cover and an exact polynomial-time algorithm for colorful edge cover, advancing solutions for these generalized covering problems.
Findings
$(2+\epsilon)$-approximation for colorful vertex cover with fixed colors
Exact $O(\omega n^3)$ algorithm for colorful edge cover
Polynomial algorithms for intermediate problems in the reduction chain
Abstract
In this paper, we study two generalizations of Vertex Cover and Edge Cover, namely Colorful Vertex Cover and Colorful Edge Cover. In the Colorful Vertex Cover problem, given an -vertex edge-colored graph with colors from and coverage requirements , the goal is to find a minimum-sized set of vertices that are incident on at least edges of color , for each , i.e., we need to cover at least edges of color . Colorful Edge Cover is similar to Colorful Vertex Cover, except here we are given a vertex-colored graph and the goal is to cover at least vertices of color , for each , by a minimum-sized set of edges. These problems have several applications in fair covering and hitting of geometric set systems involving points and lines that are divided into multiple groups.…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · Complexity and Algorithms in Graphs
