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

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.

Takehiro Ito: Supported by JSPS KAKENHI Grant Numbers 25106504, 25330003.

Naonori Kakimura: Supported by JST, ERATO, Kawarabayashi Large Graph Project, and by JSPS KAKENHI Grant Numbers 25730001, 24106002.

Naoyuki Kamiyama: Supported by JSPS KAKENHI Grant Number 24106005.

Yusuke Kobayashi: Supported by JST, ERATO, Kawarabayashi Large Graph Project, and by JSPS KAKENHI Grant Numbers 24106002, 24700004.

Yoshio Okamoto: Supported by Grant-in-Aid for Scientific Research from Ministry of Education, Science and Culture, Japan, and Japan Society for the Promotion of Science (JSPS).