IEICE TRANSACTIONS on Information
Minimum Cost Edge-Colorings of Trees Can Be Reduced to Matchings
Summary :
Let C be a set of colors, and let ω(c) be an integer cost assigned to a color c in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors in C. The cost ω(f) of an edge-coloring f of G is the sum of costs ω(f(e)) of colors f(e) assigned to all edges e in G. An edge-coloring f of G is optimal if ω(f) is minimum among all edge-colorings of G. In this paper, we show that the problem of finding an optimal edge-coloring of a tree T can be simply reduced in polynomial time to the minimum weight perfect matching problem for a new bipartite graph constructed from T. The reduction immediately yields an efficient simple algorithm to find an optimal edge-coloring of T in time O(n1.5Δlog(nNω)), where n is the number of vertices in T, Δ is the maximum degree of T, and Nω is the maximum absolute cost |ω(c)| of colors c in C. We then show that our result can be extended for multitrees.
- Publication
- IEICE TRANSACTIONS on Information Vol.E94-D No.2 pp.190-195
- Publication Date
- 2011/02/01
- Publicized
- Online ISSN
- 1745-1361
- DOI
- 10.1587/transinf.E94.D.190
- Type of Manuscript
- Special Section PAPER (Special Section on Foundations of Computer Science -- Mathematical Foundations and Applications of Algorithms and Computer Science --)
- Category
Authors
Keyword
Latest Issue
Copyrights notice of machine-translated contents
The copyright of the original papers published on this site belongs to IEICE. Unauthorized use of the original or translated papers is prohibited. See IEICE Provisions on Copyright for details.
Cite this
Copy
Takehiro ITO, Naoki SAKAMOTO, Xiao ZHOU, Takao NISHIZEKI, "Minimum Cost Edge-Colorings of Trees Can Be Reduced to Matchings" in IEICE TRANSACTIONS on Information,
vol. E94-D, no. 2, pp. 190-195, February 2011, doi: 10.1587/transinf.E94.D.190.
Abstract: Let C be a set of colors, and let ω(c) be an integer cost assigned to a color c in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors in C. The cost ω(f) of an edge-coloring f of G is the sum of costs ω(f(e)) of colors f(e) assigned to all edges e in G. An edge-coloring f of G is optimal if ω(f) is minimum among all edge-colorings of G. In this paper, we show that the problem of finding an optimal edge-coloring of a tree T can be simply reduced in polynomial time to the minimum weight perfect matching problem for a new bipartite graph constructed from T. The reduction immediately yields an efficient simple algorithm to find an optimal edge-coloring of T in time O(n1.5Δlog(nNω)), where n is the number of vertices in T, Δ is the maximum degree of T, and Nω is the maximum absolute cost |ω(c)| of colors c in C. We then show that our result can be extended for multitrees.
URL: https://global.ieice.org/en_transactions/information/10.1587/transinf.E94.D.190/_p
Copy
@ARTICLE{e94-d_2_190,
author={Takehiro ITO, Naoki SAKAMOTO, Xiao ZHOU, Takao NISHIZEKI, },
journal={IEICE TRANSACTIONS on Information},
title={Minimum Cost Edge-Colorings of Trees Can Be Reduced to Matchings},
year={2011},
volume={E94-D},
number={2},
pages={190-195},
abstract={Let C be a set of colors, and let ω(c) be an integer cost assigned to a color c in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors in C. The cost ω(f) of an edge-coloring f of G is the sum of costs ω(f(e)) of colors f(e) assigned to all edges e in G. An edge-coloring f of G is optimal if ω(f) is minimum among all edge-colorings of G. In this paper, we show that the problem of finding an optimal edge-coloring of a tree T can be simply reduced in polynomial time to the minimum weight perfect matching problem for a new bipartite graph constructed from T. The reduction immediately yields an efficient simple algorithm to find an optimal edge-coloring of T in time O(n1.5Δlog(nNω)), where n is the number of vertices in T, Δ is the maximum degree of T, and Nω is the maximum absolute cost |ω(c)| of colors c in C. We then show that our result can be extended for multitrees.},
keywords={},
doi={10.1587/transinf.E94.D.190},
ISSN={1745-1361},
month={February},}
Copy
TY - JOUR
TI - Minimum Cost Edge-Colorings of Trees Can Be Reduced to Matchings
T2 - IEICE TRANSACTIONS on Information
SP - 190
EP - 195
AU - Takehiro ITO
AU - Naoki SAKAMOTO
AU - Xiao ZHOU
AU - Takao NISHIZEKI
PY - 2011
DO - 10.1587/transinf.E94.D.190
JO - IEICE TRANSACTIONS on Information
SN - 1745-1361
VL - E94-D
IS - 2
JA - IEICE TRANSACTIONS on Information
Y1 - February 2011
AB - Let C be a set of colors, and let ω(c) be an integer cost assigned to a color c in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors in C. The cost ω(f) of an edge-coloring f of G is the sum of costs ω(f(e)) of colors f(e) assigned to all edges e in G. An edge-coloring f of G is optimal if ω(f) is minimum among all edge-colorings of G. In this paper, we show that the problem of finding an optimal edge-coloring of a tree T can be simply reduced in polynomial time to the minimum weight perfect matching problem for a new bipartite graph constructed from T. The reduction immediately yields an efficient simple algorithm to find an optimal edge-coloring of T in time O(n1.5Δlog(nNω)), where n is the number of vertices in T, Δ is the maximum degree of T, and Nω is the maximum absolute cost |ω(c)| of colors c in C. We then show that our result can be extended for multitrees.
ER -
English
