On 2-coverings and 2-packings of laminar families

Joseph Cheriyan, T. Jordán, R. Ravi

Research output: Chapter in Book/Report/Conference proceedingConference contribution

28 Citations (Scopus)

Abstract

Let H be a laminar family of subsets of a groundset V.A k-cover of H is a multiset C of edges on V such that for every subset S in H, C has at least k edges that have exactly one end in S.A k-packing of H is a multiset P of edges on V such that for every subset S in H, P has at most k • u(S) edges that have exactly one end in S. Here, u assigns an integer capacity to each subset in H. Our main results are: (a) Given a k-cover C of H, there is an efficient algorithm to find a 1-cover contained in C of size <k\C\/(2k — 1). For 2-covers, the factor of 2/3 is best possible. (b) Given a 2-packing P of H, there is an efficient algorithm to find a 1-packing contained in P of size ≥|P|/3. The factor of 1/3 for 2-packings is best possible. These results are based on efficient algorithms for finding appropriate colorings of the edges in a k-cover or a 2-packing, respectively, and they extend to the case where the edges have nonnegative weights. Our results imply approximation algorithms for some NP-hard problems in connectivity augmentation and related topics. In particular, we have a 4/3-approximation algorithm for the following problem: Given a tree T and a set of nontree edges E that forms a cycle on the leaves of T, find a minimum-size subset E' of E such that T + E' is 2-edge connected.

Original languageEnglish
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
PublisherSpringer Verlag
Pages510-520
Number of pages11
Volume1643
ISBN (Print)3540662510, 9783540662518
DOIs
Publication statusPublished - 1999
Event7th Annual European Symposium on Algorithms, ESA 1999 - Prague, Czech Republic
Duration: Jul 16 1999Jul 18 1999

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume1643
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other7th Annual European Symposium on Algorithms, ESA 1999
CountryCzech Republic
CityPrague
Period7/16/997/18/99

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

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  • Cite this

    Cheriyan, J., Jordán, T., & Ravi, R. (1999). On 2-coverings and 2-packings of laminar families. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1643, pp. 510-520). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1643). Springer Verlag. https://doi.org/10.1007/3-540-48481-7_44