A Group-Strategyproof Cost Sharing Mechanism for the Steiner Forest Game

Abstract. We consider a game-theoretical variant of the Steiner forest problem in which each player j, out of a set of k players, strives to connect his terminal pair (sj , tj) of vertices in an undirected, edge-weighted graph G. In this paper we show that a natural adaptation of the primaldual Steiner forest algorithm of Agrawal, Klein and Ravi [When trees collide: An approximation algorithm for the generalized Steiner problem in networks, SIAM Journal on Computing, 24(3):445– 456, 1995] yields a 2-budget balanced and cross-monotonic cost sharing method for this game. We also present a negative result, arguing that no cross-monotonic cost sharing method can achieve a budget balance factor of less than 2 for the Steiner tree game. This shows that our result is tight. Our algorithm gives rise to a new linear programming relaxation for the Steiner forest problem which we coin the lifted-cut relaxation. We show that this new relaxation is stronger than the standard undirected cut relaxation for the Steiner forest problem.