TY - GEN

T1 - On Fair Covering and Hitting Problems

AU - Bandyapadhyay, Sayan

AU - Banik, Aritra

AU - Bhore, Sujoy

N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - 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, …, ω} and coverage requirements r1, r2, …, rω, the goal is to find a minimum-sized set of vertices that are incident on at least ri edges of color i, for each 1 ≤ i≤ ω, i.e., we need to cover at least ri 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 ri vertices of color i, for each 1 ≤ i≤ ω, 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 + ϵ) -approximation for the Colorful Vertex Cover problem in time nO ( ω / ϵ ), i.e., we obtain an O(1)-approximation in polynomial time for constant number of colors. Next, for the Colorful Edge Cover problem, we design an O(ωn3) 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.

AB - 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, …, ω} and coverage requirements r1, r2, …, rω, the goal is to find a minimum-sized set of vertices that are incident on at least ri edges of color i, for each 1 ≤ i≤ ω, i.e., we need to cover at least ri 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 ri vertices of color i, for each 1 ≤ i≤ ω, 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 + ϵ) -approximation for the Colorful Vertex Cover problem in time nO ( ω / ϵ ), i.e., we obtain an O(1)-approximation in polynomial time for constant number of colors. Next, for the Colorful Edge Cover problem, we design an O(ωn3) 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.

UR - http://www.scopus.com/inward/record.url?scp=85115877002&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-86838-3_4

DO - 10.1007/978-3-030-86838-3_4

M3 - Conference contribution

AN - SCOPUS:85115877002

SN - 9783030868376

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 39

EP - 51

BT - Graph-Theoretic Concepts in Computer Science - 47th International Workshop, WG 2021, Revised Selected Papers

A2 - Kowalik, Lukasz

A2 - Pilipczuk, Michal

A2 - Rzazewski, Pawel

PB - Springer Science and Business Media Deutschland GmbH

T2 - 47th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2021

Y2 - 23 June 2021 through 25 June 2021

ER -