Minimum-Cost b-Edge Dominating Sets on Trees

Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, Yoshio Okamoto

Research output: Contribution to journalArticle

Abstract

In this paper, we consider the minimum-cost b-edge dominating set problem. This is a generalization of the edge dominating set problem, but the computational complexity for trees is an astonishing open problem. We make steps toward the resolution of this open problem in the following three directions. (1) We give the first combinatorial polynomial-time algorithm for paths. Prior to our work, the polynomial-time algorithm for paths used linear programming, and it was known that the linear-programming approach could not be extended to trees. Thus, our algorithm would yield an alternative approach to a possible polynomial-time algorithm for trees. (2) We give a fixed-parameter algorithm for trees with the number of leaves as a parameter. Thus, a possible NP-hardness proof for trees should make use of trees with unbounded number of leaves. (3) We give a fully polynomial-time approximation scheme for trees. Prior to our work, the best known approximation factor was two. If the problem is NP-hard, then a possible proof cannot be done via a gap-preserving reduction from any APX-hard problem unless P(Formula presented.)NP.

Original languageEnglish
Pages (from-to)1-24
Number of pages24
JournalAlgorithmica
DOIs
Publication statusAccepted/In press - 2018 May 10

Fingerprint

Dominating Set
Polynomials
Costs
Polynomial-time Algorithm
Linear programming
Computational complexity
Open Problems
Leaves
Trees (mathematics)
Fully Polynomial Time Approximation Scheme
Fixed-parameter Algorithms
Path
Combinatorial Algorithms
NP-hardness
Hardness
Best Approximation
Computational Complexity
NP-complete problem
Alternatives

Keywords

  • Dynamic programming
  • Edge dominating set
  • Tree

ASJC Scopus subject areas

  • Computer Science(all)
  • Computer Science Applications
  • Applied Mathematics

Cite this

Ito, T., Kakimura, N., Kamiyama, N., Kobayashi, Y., & Okamoto, Y. (Accepted/In press). Minimum-Cost b-Edge Dominating Sets on Trees. Algorithmica, 1-24. https://doi.org/10.1007/s00453-018-0448-z

Minimum-Cost b-Edge Dominating Sets on Trees. / Ito, Takehiro; Kakimura, Naonori; Kamiyama, Naoyuki; Kobayashi, Yusuke; Okamoto, Yoshio.

In: Algorithmica, 10.05.2018, p. 1-24.

Research output: Contribution to journalArticle

Ito, Takehiro ; Kakimura, Naonori ; Kamiyama, Naoyuki ; Kobayashi, Yusuke ; Okamoto, Yoshio. / Minimum-Cost b-Edge Dominating Sets on Trees. In: Algorithmica. 2018 ; pp. 1-24.
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