Minimum cost edge-colorings of trees can be reduced to matchings

Takehiro Ito, Naoki Sakamoto, Xiao Zhou, Takao Nishizeki

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 , where n is the number of vertices in T, Δ is the maximum degree of T, and N ω is the maximum absolute cost |ω(c)| of colors c in C. We then show that our result can be extended for multitrees.

Original languageEnglish
Title of host publicationFrontiers in Algorithmics - 4th International Workshop, FAW 2010, Proceedings
Pages274-284
Number of pages11
DOIs
Publication statusPublished - 2010 Aug 27
Event4th International Frontiers of Algorithmics Workshop, FAW 2010 - Wuhan, China
Duration: 2010 Aug 112010 Aug 13

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6213 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other4th International Frontiers of Algorithmics Workshop, FAW 2010
CountryChina
CityWuhan
Period10/8/1110/8/13

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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    Ito, T., Sakamoto, N., Zhou, X., & Nishizeki, T. (2010). Minimum cost edge-colorings of trees can be reduced to matchings. In Frontiers in Algorithmics - 4th International Workshop, FAW 2010, Proceedings (pp. 274-284). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6213 LNCS). https://doi.org/10.1007/978-3-642-14553-7_26