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Minimum-Cost \(b\)-Edge Dominating Sets on Trees

  • Takehiro Ito
  • Naonori Kakimura
  • Naoyuki Kamiyama
  • Yusuke Kobayashi
  • Yoshio Okamoto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8889)

Abstract

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 \(=\) NP.

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References

  1. 1.
    Berger, A., Fukunaga, T., Nagamochi, H., Parekh, O.: Approximability of the capacitated-edge dominating set problem. Theor. Comput. Sci. 385(1–3), 202–213 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Berger, A., Parekh, O.: Linear time algorithms for generalized edge dominating set problems. Algorithmica 50(2), 244–254 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Berger, A., Parekh, O.: Erratum to: Linear Time Algorithms for Generalized Edge Dominating Set Problems. Algorithmica 62(1), 633–634 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dadush, D., Peikert, C., Vempala, S.: Enumerative lattice algorithms in any norm via M-ellipsoid coverings. In: FOCS, pp. 580–589 (2011)Google Scholar
  5. 5.
    Dadush, D., Vempala, S.: Deterministic construction of an approximate M-ellipsoid and its applications to derandomizing lattice algorithms. In: SODA, pp. 1445–1456 (2012)Google Scholar
  6. 6.
    Fujito, T., Nagamochi, H.: A \(2\)-approximation algorithm for the minimum weight edge dominating set problem. Discrete Appl. Math. 118(3), 199–207 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer (2004)Google Scholar
  8. 8.
    Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Parekh, O.: Edge dominating and hypomatchable sets. In: SODA, 287–291 (2002)Google Scholar
  10. 10.
    Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons (1986)Google Scholar
  11. 11.
    Tardos, É.: A strongly polynomial algorithm to solve combinatorial linear programs. Operations Research 34(2), 250–256 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Vazirani, V.V.: Approximation algorithms. Springer (2001)Google Scholar
  13. 13.
    Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38(3), 364–372 (1980)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Takehiro Ito
    • 1
  • Naonori Kakimura
    • 2
  • Naoyuki Kamiyama
    • 3
  • Yusuke Kobayashi
    • 2
  • Yoshio Okamoto
    • 4
  1. 1.Tohoku UniversitySendaiJapan
  2. 2.University of TokyoTokyoJapan
  3. 3.Kyushu UniversityFukuokaJapan
  4. 4.University of Electro-CommunicationsTokyoJapan

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