Parameterized Complexity of d-Hitting Set with Quotas

Sushmita Gupta, Pallavi Jain, Aditya Petety, Sagar Singh

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

In this paper we study a variant of the classic d -Hitting Set problem with lower and upper capacity constraints, say A and B, respectively. The input to the problem consists of a universe U, a set family, S, of sets over U, where each set in the family is of size at most d, a non-negative integer k; and additionally two functions α: S→ { 1, …, A} and β: S→ { 1, …, B}. The goal is to decide if there exists a hitting set of size at most k such that for every set S in the family S, the solution contains at least α(S) elements and at most β(S) elements from S. We call this the (A, B) -Multi d-Hitting Set problem. We study the problem in the realm of parameterized complexity. We show that (A, B) -Multi d-Hitting Set can be solved in O(dk) time. For the special case when d= 3 and d= 4, we have an improved bound of O(2. 2738 k) and O(3. 562 k), respectively. The former matches the running time of the classical 3-Hitting Set problem. Furthermore, we show that if we do not have an upper bound constraint and the lower bound constraint is same for all the sets in the family, say A> 1, then the problem can be solved even faster than d-Hitting Set. We next investigate some graph-theoretic problems which can be thought of as an implicit d-Hitting Set problem. In particular, we study (A, B) -Multi Vertex Cover and (A, B) -Multi Feedback Vertex Set in Tournaments. In (A, B) -Multi Vertex Cover, we are given a graph G and a non-negative integer k, the goal is to find a subset S⊆ V(G) of size at most k such that for every edge in G, S contains at least A and at most B of its endpoints. Analogously, we can define (A, B) -Multi Feedback Vertex Set in Tournaments. We show that unlike Vertex Cover, which is same as (1, 2 ) -Multi Vertex Cover, (1, 1 ) -Multi Vertex Cover is polynomial-time solvable. Furthermore, unlike Feedback Vertex Set in Tournaments, (A, B) -Multi Feedback Vertex Set in Tournaments can be solved in polynomial time.

Original languageEnglish
Title of host publicationSOFSEM 2021
Subtitle of host publicationTheory and Practice of Computer Science - 47th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2021, Proceedings
EditorsTomáš Bureš, Riccardo Dondi, Johann Gamper, Giovanna Guerrini, Tomasz Jurdzinski, Claus Pahl, Florian Sikora, Prudence W. Wong
PublisherSpringer Science and Business Media Deutschland GmbH
Pages293-307
Number of pages15
ISBN (Print)9783030677305
DOIs
StatePublished - 1 Jan 2021
Externally publishedYes
Event47th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2021 - Bolzano-Bozen, Italy
Duration: 25 Jan 202129 Jan 2021

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12607 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference47th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2021
Country/TerritoryItaly
CityBolzano-Bozen
Period25/01/2129/01/21

Keywords

  • Kernelization
  • Multi hitting set
  • Parameterized complexity

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science

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