### 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.

Original language | English |
---|---|

Pages (from-to) | 343-366 |

Number of pages | 24 |

Journal | Algorithmica |

Volume | 81 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 15 2019 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Algorithmica*,

*81*(1), 343-366. 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.

Research output: Contribution to journal › Article

*Algorithmica*, vol. 81, no. 1, pp. 343-366. https://doi.org/10.1007/s00453-018-0448-z

}

TY - JOUR

T1 - Minimum-Cost b-Edge Dominating Sets on Trees

AU - Ito, Takehiro

AU - Kakimura, Naonori

AU - Kamiyama, Naoyuki

AU - Kobayashi, Yusuke

AU - Okamoto, Yoshio

PY - 2019/1/15

Y1 - 2019/1/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85046826637&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046826637&partnerID=8YFLogxK

U2 - 10.1007/s00453-018-0448-z

DO - 10.1007/s00453-018-0448-z

M3 - Article

AN - SCOPUS:85046826637

VL - 81

SP - 343

EP - 366

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 1

ER -