An efficient cost-sharing mechanism for the prize-collecting steiner forest problem

In an instance of the prize-collecting Steiner forest problem (PCSF) we are given an undirected graph G = (V,E), non-negative edge-costs c(e) for all e ∈ E, terminal pairs R = {(si, ti)}1≤i≤k, and penalties π1, ⋯, π k. A feasible solution (F,Q) consists of a forest F and a subset Q of terminal pairs such that for all (si, ti) ∈ R either si, ti are connected by F or (si, ti) ∈ Q. The objective is to compute a feasible solution of minimum cost c(F) + π(Q). A game-theoretic version of the above problem has k players, one for each terminal-pair in R. Player i's ultimate goal is to connect si and ti, and the player derives a privately held utility ui ≥ 0 from being connected. A service provider can connect the terminals si and ti of player i in two ways: (1) by buying the edges of an si, ti-path in G, or (2) by buying an alternate connection between si and ti (maybe from some other provider) at a cost of πi. In this paper, we present a simple 3-budget-balanced and group-strategyproof mechanism for the above problem. We also show that our mechanism computes client sets whose social cost is at most O(log2 k) times the minimum social cost of any player set. This matches a lower-bound that was recently given by Roughgarden and Sundararajan (STOC '06). Copyright © 2007 by the Association for Computing Machinery, Inc. and the Society for Industrial and Applied Mathematics.