## 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**.

## Keywords

Edge dominating set Tree Dynamic programming## Notes

### 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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