On Euclidean Prize-collecting Steiner Forest Problems

In this paper, we consider prize-collecting Steiner forest and k-Steiner forest when the vertices of the input graph are points in the Euclidean plane and the lengths are Euclidean distances. First, we simplify the analysis of the polynomial-time approximation scheme (PTAS) of Borradaile et al. [7, 8] for the Euclidean Steiner forest problem. This is done by proving a new structural property and slightly modifying the dynamic programming by adding a new piece of information to each dynamic programming state (we do believe it is necessary but missing in the implicit dynamic programming of Borradaile et al.). Then based on our new structural property and algorithm, we develop a PTAS for a well-motivated case, i.e., the multiplicative case, of prize-collecting and budgeted Steiner forest. At the end, we present some evidence that why these problems might be hard if they are in the general (and not multiplicative) Euclidean case. ∗Department of Computer Science, Princeton University, Princeton, NJ 08540; Email: [email protected]. †AT&T Labs– Research, Florham Park, NJ 07932; Email: [email protected].