On the complexity of constructing minimum changeover cost arborescences

The reload cost concept refers to the cost that occurs at a vertex along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. This cost depends only on the colors of the traversed edges. Reload costs arise in various applications such as transportation networks, energy distribution networks, and telecommunications. Previous work on reload costs focuses on two problems of finding a spanning tree with minimum cost with respect to two different cost measures. In both problems the cost is associated with a set of paths from a given vertex r to all the leaves of the constructed tree. The first cost measure is the sum of the reload costs of all paths from r to the leaves. The second cost measure is the changeover cost, in which the cost of traversing a vertex by using two specific incident edges is paid only once regardless of the number of paths traversing it. The first problem is inapproximable within any polynomial time computable function of the input size [1], and the second problem is inapproximable within n1−ϵ for any ϵ>0 [2]. In this paper we show that the first hardness result holds also for the second problem. Given this strong inapproximability result, we study the complexity and approximability properties of numerous special cases of this second problem. We mainly focus on bounded costs, and consider both directed and undirected graphs, bounded and unbounded number of colors, and both bounded and unbounded degree graphs. We also present polynomial time exact algorithms and an approximation algorithm for some special case. To the best of our knowledge, these are the first algorithms with a provable performance guarantee for the problem. Moreover, our approximation algorithm shows a tight bound on the approximability of the problem for a specific family of instances.