Minimum Reload Cost Cycle Cover in Complete Graphs [article]

Yasemin Büyükçolak, Didem Gözüpek, Sibel Özkan
2017 arXiv   pre-print
The reload cost refers to the cost that occurs along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. Galbiati et al.[1] introduced the Minimum Reload Cost Cycle Cover problem, which is to find a set of vertex-disjoint cycles spanning all vertices with minimum reload cost. They proved that this problem is strongly NP-hard and not approximable within 1/ϵ for any ϵ > 0 even when the number of colors is 2, the reload costs are symmetric
more » ... satisfy the triangle inequality. In this paper, we study this problem in complete graphs having equitable or nearly equitable 2-edge-colorings, which are edge-colorings with two colors such that for each vertex v ∈ V(G), ||c_1(v)| -|c_2(v)|| ≤ 1 or ||c_1(v)| -|c_2(v)|| ≤ 2, respectively, where c_i(v) is the set of edges with color i that is incident to v. We prove that except possibly on complete graphs with fewer than 13 vertices, the minimum reload cost is zero on complete graphs with nearly equitable 2-edge-colorings by proving the existence of a monochromatic cycle cover. Furthermore, we provide a polynomial-time algorithm that constructs a monochromatic cycle cover in complete graphs with an equitable 2-edge-coloring except possibly in a complete graph with four vertices. Our algorithm also finds a monochromatic cycle cover in complete graphs with a nearly equitable 2-edge-coloring except some special cases.
arXiv:1706.05225v1 fatcat:xm3vmczg4fapxp7rq3er2zeztq