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,ti) 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. © Springer-Verlag Berlin Heidelberg 2005.
CITATION STYLE
Könemann, J., Leonardi, S., Schäfer, G., & Van Zwam, S. (2005). From primal-dual to cost shares and back: A stronger LP relaxation for the Steiner forest problem. In Lecture Notes in Computer Science (Vol. 3580, pp. 930–942). Springer Verlag. https://doi.org/10.1007/11523468_75
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