Approximation algorithm for steiner tree problem with neighbor-induced cost

The Steiner tree problem is one of the most fundamental combinatorial optimization problems. In this problem, the input is an undirected graph, a cost function on the edge set, and a subset of vertices called terminals, and the objective is to find a minimum-cost tree spanning all the terminals. By changing the objective function, several variants of the Steiner tree problem have been studied in the literature. For example, in the min-power version of the problem, the goal is to find a Steiner tree S minimizing the total power consumption of vertices, where the power of a vertex v is the maximum cost of any edge of Sincident to v. Another example is the node-weighted version of the problem, in which costs are assigned to vertices instead of edges. In this paper, we introduce a common generalization of these variants and give an approximation algorithm for it. When the maximum degree Δ of the input graph is constant, our algorithm runs in polynomial time and the approximation ratio is 2.016 + ln Δ.