Mathematical Programming

, Volume 152, Issue 1–2, pp 147–188 | Cite as

Efficient cost-sharing mechanisms for prize-collecting problems

  • A. Gupta
  • J. Könemann
  • S. Leonardi
  • R. Ravi
  • G. Schäfer
Full Length Paper Series A

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.

Mathematics Subject Classification

90C27 68R99 91A10 

Notes

Acknowledgments

We thank anonymous reviewers of Mathematical Programming for their valuable comments and most helpful feedback.

References

  1. 1.
    Agrawal, A., Klein, P., Ravi, R.: When trees collide: an approximation algorithm for the generalized Steiner problem on networks. SIAM J. Comput. 24(3), 440–456 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Archer, A., Bateni, M., Hajiaghayi, M., Karloff, H.: Improved approximation algorithms for prize-collecting steiner tree and TSP. SIAM J. Comput. 40(2), 309–332 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Archer, A., Feigenbaum, J., Krishnamurthy, A., Sami, R., Shenker, S.: Approximation and collusion in multicast cost sharing. Games Econ. Behav. 47(1), 36–71 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. J. ACM 45(3), 501–555 (1998)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bartal, Y.: Probabilistic approximations of metric spaces and its algorithmic applications. In: Proceedings of the 37th Symposium on the Foundations of Computer, Science, pp. 184–193 (1996)Google Scholar
  6. 6.
    Bern, M., Plassman, P.: The Steiner problem with edge lengths 1 and 2. Inf. Process. Lett. 32, 171–176 (1989)CrossRefMATHGoogle Scholar
  7. 7.
    Bienstock, D., Goemans, M.X., Simchi-Levi, D., Williamson, D.P.: A note on the prize collecting traveling salesman problem. Math. Program. 59, 413–420 (1993)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bleischwitz, Y., Monien, B.: Fair cost-sharing methods for scheduling jobs on parallel machines. J. Discrete Algorithms 7(3), 280–290 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bleischwitz, Y., Monien, B., Schoppmann, F.: To be or not to be (served). In: Proceedings of the 3rd International Conference on Internet and Network Economics, pp. 515–528 (2007)Google Scholar
  10. 10.
    Brenner, J., Schäfer, G.: Cost sharing methods for makespan and completion time scheduling. In: Proceedings of the 24th International Symposium on Theoretical Aspects of Computer, Science, pp. 670–681 (2007)Google Scholar
  11. 11.
    Chawla, S., Roughgarden, T., Sundararajan, M.: Optimal cost-sharing mechanisms for steiner forest problems. In: Proceedings of the 2nd International Workshop on Internet and Network Economics, pp. 112–123 (2006)Google Scholar
  12. 12.
    Dutta, B., Ray, D.: A concept of egalitarianism under participation constraints. Econometrica 57(3), 615–635 (1989)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci. 69(3), 485–497 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Feigenbaum, J., Krishnamurthy, A., Sami, R., Shenker, S.: Hardness results for multicast cost-sharing. Theor. Comput. Sci. 304, 215–236 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Feigenbaum, J., Papadimitriou, C.H., Shenker, S.: Sharing the cost of multicast transmissions. J. Comput. Syst. Sci. 63(1), 21–41 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24(2), 296–317 (1995)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Green, J., Kohlberg, E., Laffont, J.J.: Partial equilibrium approach to the free rider problem. J. Public Econ. 6, 375–394 (1976)CrossRefGoogle Scholar
  18. 18.
    Gupta, A., Srinivasan, A., Tardos, É.: Cost-sharing mechanisms for network design. Algorithmica 50(1), 98–119 (2008)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hajiaghayi, M.T., Jain, K.: The prize-collecting generalized Steiner tree problem via a new approach of primal-dual schema. In: Proceedings of the 17th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 631–640 (2006)Google Scholar
  20. 20.
    Immorlica, N., Mahdian, M., Mirrokni, V.S.: Limitations of cross-monotonic cost sharing schemes. In: Proceedings of the 16th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 602–611 (2005)Google Scholar
  21. 21.
    Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM 48(4), 761–777 (2001)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jain, K., Vazirani, V.: Applications of approximation algorithms to cooperative games. In: Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing, pp. 364–372 (2001)Google Scholar
  23. 23.
    Jain, K., Vazirani, V.V.: Equitable cost allocations via primal-dual-type algorithms. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pp. 313—321 (2002)Google Scholar
  24. 24.
    Kent, K.J., Skorin-Kapov, D.: Population monotonic cost allocations on MSTs. In: Proceedings of the 6th International Conference on Operational Research, pp. 43–48 (1996)Google Scholar
  25. 25.
    Könemann, J., Leonardi, S., Schäfer, G., van Zwam, S.: From primal-dual to cost shares and back: a stronger LP relaxation for the Steiner forest problem. In: Proceedings of the 32nd International Colloquium on Automata, Languages and Programming, pp. 930–942 (2005)Google Scholar
  26. 26.
    Könemann, J., Leonardi, S., Schäfer, G., van Zwam, S.H.M.: A group-strategyproof cost sharing mechanism for the Steiner forest game. SIAM J. Comput. 37(5), 1319–1341 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Leonardi, S., Schäfer, G.: Cross-monotonic cost sharing methods for connected facility location games. Theoret. Comput. Sci. 326(1–3), 431–442 (2004)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Mettu, R.R., Plaxton, C.G.: The online median problem. SIAM J. Comput. 32(3), 816–832 (2003)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Moulin, H.: Incremental cost sharing: characterization by coalition strategy-proofness. Soc. Choice Welfare 16, 279–320 (1999)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Moulin, H., Shenker, S.: Strategyproof sharing of submodular costs: budget balance versus efficiency. Econ. Theor. 18(3), 511–533 (2001)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V. (eds.).: Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)Google Scholar
  32. 32.
    Pál, M., Tardos, E.: Group strategyproof mechanisms via primal-dual algorithms. In: Proceedings of the 44th Symposium on the Foundations of Computer, Science, pp. 584–593 (2003)Google Scholar
  33. 33.
    Pountourakis, E., Vidali, A.: A complete characterization of group-strategyproof mechanisms of cost-sharing. In: Proceedings of the 18th Annual European Symposium on Algorithms, pp. 146–157 (2010)Google Scholar
  34. 34.
    Roberts, K.: The characterization of implementable choice rules. In: Laffont, J.J. (ed.) Aggregation and Revelation of Preferences. North-Holland, Amsterdam (1979)Google Scholar
  35. 35.
    Roughgarden, T., Sundararajan, M.: Optimal efficiency guarantees for network design mechanisms. In: Proceedings of the 12th International Conference on Integer Programming and Combinatorial Optimization, pp. 469–483 (2007)Google Scholar
  36. 36.
    Roughgarden, T., Sundararajan, M.: Quantifying inefficiency in cost-sharing mechanisms. J. ACM 56(4), 1–33 (2009)Google Scholar
  37. 37.
    Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory Ser. B 80(2), 346–355 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  • A. Gupta
    • 1
  • J. Könemann
    • 2
  • S. Leonardi
    • 3
  • R. Ravi
    • 4
  • G. Schäfer
    • 5
    • 6
  1. 1.School of Computer ScienceCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  3. 3.Department of Computer, Control and Management Engineering ‘Antonio Ruberti’Sapienza University of RomeRomeItaly
  4. 4.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA
  5. 5.Networks and Optimization GroupCentrum Wiskunde & Informatica (CWI)AmsterdamThe Netherlands
  6. 6.Department of Econometrics and Operations ResearchVU University AmsterdamAmsterdamThe Netherlands

Personalised recommendations