TY - GEN

T1 - Minimum cost edge-colorings of trees can be reduced to matchings

AU - Ito, Takehiro

AU - Sakamoto, Naoki

AU - Zhou, Xiao

AU - Nishizeki, Takao

PY - 2010

Y1 - 2010

N2 - 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.

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 , 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.

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U2 - 10.1007/978-3-642-14553-7_26

DO - 10.1007/978-3-642-14553-7_26

M3 - Conference contribution

AN - SCOPUS:77955882397

SN - 3642145523

SN - 9783642145520

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 274

EP - 284

BT - Frontiers in Algorithmics - 4th International Workshop, FAW 2010, Proceedings

T2 - 4th International Frontiers of Algorithmics Workshop, FAW 2010

Y2 - 11 August 2010 through 13 August 2010

ER -