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 Open image in new window , 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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