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

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(n^1.5Δ log(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.