Abstract
We consider the problem of designing efficient mechanisms to share the cost of providing some service to a set of self-interested customers. In this paper, we mainly focus on cost functions that are induced by prize-collecting optimization problems. Such cost functions arise naturally whenever customers can be served in two different ways: either by being part of a common service solution or by being served individually. One of our main contributions is a general lifting technique that allows us to extend the social cost approximation guarantee of a Moulin mechanism for the respective non-prize-collecting problem to its prize-collecting counterpart. Our lifting technique also suggests a generic design template to derive Moulin mechanisms for prize-collecting problems. The approach is particularly suited for cost-sharing methods that are based on primal-dual algorithms. We illustrate the applicability of our approach by deriving Moulin mechanisms for prize-collecting variants of submodular cost-sharing, facility location and Steiner forest problems. All our mechanisms are essentially best possible with respect to budget balance and social cost approximation guarantees. Finally, we show that the Moulin mechanism by Könemann et al. (SIAM J Comput 37(5):1319–1341, 2008) for the Steiner forest problem is \(O(\log ^3 k)\)-approximate. Our approach adds a novel methodological contribution to existing techniques by showing that such a result can be proved by embedding the graph distances into random hierarchically separated trees.
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Notes
An equivalent definition of submodularity is that \(c\) has non-increasing marginal costs, i.e., for every player \(i \in U\) and every set \(S \subseteq T \subseteq U{\setminus } \{i\}\), \(c(S \cup \{i\}) - c(S) \ge c(T \cup \{i\}) - c(T)\).
This follows from Theorem 4 in [9] and the observation that for submodular cost functions their egalitarian mechanism coincides with the Moulin mechanism \(M(\xi ^\mathtt{SC })\).
The latter initialization is only done implicitly. The algorithm will ensure that the cuts with positive dual value form a laminar family.
Note that in PSF the decision of which edges are added to \(F^\tau \) are delayed until these edges are non-redundant. This is different from other primal-dual approaches for network design problems (see, e.g., [16]) where this is ensured by a final reverse-delete step.
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We thank anonymous reviewers of Mathematical Programming for their valuable comments and most helpful feedback.
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A preliminary version of this paper appeared under the title “An efficient cost-sharing mechanism for the prize-collecting Steiner forest problem” in the Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1153–1162, ACM Press, 2007.
J. Könemann received the Research support by NSERC grant no. 288340-2004 and by an IBM Faculty Award.
Part of this work was done while S. Leonardi was visiting the School of Computer Science at Carnegie Mellon University.
Part of this work was done while G. Schäfer was visiting the School of Computer Science at Carnegie Mellon University.
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Gupta, A., Könemann, J., Leonardi, S. et al. Efficient cost-sharing mechanisms for prize-collecting problems. Math. Program. 152, 147–188 (2015). https://doi.org/10.1007/s10107-014-0781-1
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DOI: https://doi.org/10.1007/s10107-014-0781-1