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Minimum-Cost b-Edge Dominating Sets on Trees

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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\(=\)NP.

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Notes

  1. In [2], the authors claimed that b-eds on trees can be solved in polynomial time via linear programming. However this claim is not correct (see [3]).

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Acknowledgements

The authors would like to thank Yusuke Matsumoto and Chien-Chung Huang for helpful discussions on this topic. The authors would like to thank anonymous referees of ISAAC2014 and of this journal version for helpful comments. Takehiro Ito was partially supported by JST CREST Grant Number JPMJCR1402 and JSPS KAKENHI Grant Numbers JP25330003, JP15H00849, and JP16K00004. Naonori Kakimura was partially supported by JST ERATO Grant Number JPMJER1201 and JSPS KAKENHI Grant Numbers JP25730001 and JP24106002. Naoyuki Kamiyama was supported by JSPS KAKENHI Grant Number JP24106005. Yusuke Kobayashi was supported by JST ERATO Grant Number JPMJER1201 and JSPS KAKENHI Grant Numbers JP24106002 and JP24700004. Yoshio Okamoto was partially supported by JST CREST Grant Number JPMJCR1402 and JSPS/MEXT Grant-in-Aid for Scientific Research Grant Number JP24106005, JP24700008, JP24220003, and JP15K00009.

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Authors and Affiliations

  1. Tohoku University, Sendai, Japan

    Takehiro Ito

  2. Keio University, Yokohama, Japan

    Naonori Kakimura

  3. Kyushu University, Fukuoka, Japan

    Naoyuki Kamiyama

  4. University of Tsukuba, Tsukuba, Japan

    Yusuke Kobayashi

  5. University of Electro-Communications, Chofu, Japan

    Yoshio Okamoto

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  1. Takehiro Ito
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  2. Naonori Kakimura
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  3. Naoyuki Kamiyama
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  4. Yusuke Kobayashi
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  5. Yoshio Okamoto
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Corresponding author

Correspondence to Naoyuki Kamiyama.

Additional information

An earlier version of this paper has appeared in Proceedings of the 25th International Symposium on Algorithms and Computation (ISAAC), Lecture Notes in Computer Science 8889, pp. 195–207 (2014).

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Ito, T., Kakimura, N., Kamiyama, N. et al. Minimum-Cost b-Edge Dominating Sets on Trees. Algorithmica 81, 343–366 (2019). https://doi.org/10.1007/s00453-018-0448-z

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