Hitting sets online and vertex ranking

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

1 Scopus citations

Abstract

We consider the problem of hitting sets online for several families of hypergraphs. Given a graph G = (V,E), let H = (V,R) denote the hypergraph whose hyperedges are the subsets U ∈ V such that the induced subgraph G[U] is connected. We establish a new connection between the best competitive ratio for the online hitting set problem in H and the vertex ranking number of G. This connection states that these two parameters are equal. Moreover, this equivalence is algorithmic in the sense, that given an algorithm to compute a vertex ranking of G with k colors, one can use this algorithm as a black-box in order to design a k-competitive deterministic online hitting set algorithm for H. Also, given a deterministic k-competitive online algorithm for H, we can use it as a black box in order to compute a vertex ranking for G with at most k colors. As a corollary, we obtain optimal online hitting set algorithms for many such hypergraphs including those realized by planar graphs, graphs with bounded tree width, trees, etc. This improves the best previously known and more general bound of Alon et al. We also consider two geometrically defined hypergraphs. The first one is defined by subsets of a given set of n points in the Euclidean plane that are induced by half-planes. We obtain an O(logn)-competitive ratio. We also prove an Ω(log n) lower bound for the competitive ratio in this setting. The second hypergraph is defined by subsets of a given set of n points in the plane induced by unit discs. Since the number of subsets in this setting is O(n2), the competitive ratio obtained by Alon et al. is O(log2n). We introduce an algorithm with O(log n)-competitive ratio. We also show that any online algorithm for this problem has a competitive ratio of Ω(log n), and hence our algorithm is optimal.

Original languageEnglish
Title of host publicationAlgorithms, ESA 2011 - 19th Annual European Symposium, Proceedings
Pages347-357
Number of pages11
DOIs
StatePublished - 20 Sep 2011
Event19th Annual European Symposium on Algorithms, ESA 2011 - Saarbrucken, Germany
Duration: 5 Sep 20119 Sep 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6942 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference19th Annual European Symposium on Algorithms, ESA 2011
Country/TerritoryGermany
CitySaarbrucken
Period5/09/119/09/11

Keywords

  • Geometric Hypergraphs
  • Hitting Set
  • Online Algorithms
  • Set Cover

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)

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