On Colorful Vertex and Edge Cover Problems

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 $n$-vertex edge-colored graph $G$ with colors from $\{1, \ldots, \omega\}$ and coverage requirements $r_1, r_2, \ldots, r_\omega$, the goal is to find a minimum-sized set of vertices that are incident on at least $r_i$ edges of color $i$, for each $1 \le i \le \omega$, i.e., we need to cover at least $r_i$ edges of color $i$. 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 $r_i$ vertices of color $i$, for each $1 \le i \le \omega$, 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. Here, fairness ensures that the coverage (resp. hitting) requirement of every group is fully satisfied. We obtain a $(2+\epsilon)$-approximation for the Colorful Vertex Cover problem in time $n^{O(\omega/\epsilon)}$. Thus, for a constant number of colors, the problem admits a $(2+\epsilon)$-approximation in polynomial time. Next, for the Colorful Edge Cover problem, we design an $O(\omega n^3)$ time exact algorithm, via a chain of reductions to a matching problem. For all intermediate problems in this chain of reductions, we design polynomial-time algorithms, which might be of independent interest.