From primal-dual to cost shares and back: a stronger LP relaxation for the Steiner forest problem

We consider a game-theoretical variant of the Steiner forest problem, in which each of k users i strives to connect his terminal pair (s(i), t(i)) of vertices in an undirected, edge-weighted graph G. In [1] a natural primal-dual algorithm was shown which achieved a 2-approximate budget balanced cross-monotonic cost sharing method for this game. We derive a new linear programming relaxation from the techniques of [1] which allows for a simpler proof of the budget balancedness of the algorithm from [1]. Furthermore we show that this new relaxation is strictly stronger than the well-known undirected cut relaxation for the Steiner forest problem. We conclude the paper with 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 and Steiner forest games. This shows that the results of [1, 2] are essentially tight.