Advertisement

On Budget-Feasible Mechanism Design for Symmetric Submodular Objectives

  • Georgios Amanatidis
  • Georgios Birmpas
  • Evangelos Markakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10660)

Abstract

We study a class of procurement auctions with a budget constraint, where an auctioneer is interested in buying resources from a set of agents. The auctioneer would like to select a subset of the resources so as to maximize his valuation function, without exceeding his budget. As the resources are owned by strategic agents, our overall goal is to design mechanisms that are truthful, budget-feasible, and obtain a good approximation to the optimal value. Previous results on budget-feasible mechanisms have considered mostly monotone valuation functions. In this work, we mainly focus on symmetric submodular valuations, a prominent class of non-monotone submodular functions that includes cut functions. We begin with a purely algorithmic result, obtaining a \(\frac{2e}{e-1}\)-approximation for maximizing symmetric submodular functions under a budget constraint. We then proceed to propose truthful, budget feasible mechanisms (both deterministic and randomized), paying particular attention on the Budgeted Max Cut problem. Our results significantly improve the known approximation ratios for these objectives, while establishing polynomial running time for cases where only exponential mechanisms were known. At the heart of our approach lies an appropriate combination of local search algorithms with results for monotone submodular valuations, applied to the derived local optima.

References

  1. 1.
    Ageev, A.A., Sviridenko, M.I.: Approximation algorithms for maximum coverage and max cut with given sizes of parts. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds.) IPCO 1999. LNCS, vol. 1610, pp. 17–30. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48777-8_2 CrossRefGoogle Scholar
  2. 2.
    Amanatidis, G., Birmpas, G., Markakis, E.: Coverage, matching, and beyond: new results on budgeted mechanism design. In: Cai, Y., Vetta, A. (eds.) WINE 2016. LNCS, vol. 10123, pp. 414–428. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-54110-4_29 CrossRefGoogle Scholar
  3. 3.
    Amanatidis, G., Birmpas, G., Markakis, E.: On budget-feasible mechanism design for symmetric submodular objectives. CoRR abs/1704.06901 (2017)Google Scholar
  4. 4.
    Anari, N., Goel, G., Nikzad, A.: Mechanism design for crowdsourcing: an optimal 1-1/e competitive budget-feasible mechanism for large markets. In: 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, pp. 266–275 (2014)Google Scholar
  5. 5.
    Bei, X., Chen, N., Gravin, N., Lu, P.: Budget feasible mechanism design: from prior-free to Bayesian. In: Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, pp. 449–458 (2012)Google Scholar
  6. 6.
    Borodin, A., Filmus, Y., Oren, J.: Threshold models for competitive influence in social networks. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 539–550. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-17572-5_48 CrossRefGoogle Scholar
  7. 7.
    Caselton, W.F., Zidek, J.V.: Optimal monitoring network designs. Stat. Probab. Lett. 2(4), 223–227 (1984)CrossRefMATHGoogle Scholar
  8. 8.
    Chekuri, C., Vondrák, J., Zenklusen, R.: Submodular function maximization via the multilinear relaxation and contention resolution schemes. SIAM J. Comput. 43(6), 1831–1879 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chen, N., Gravin, N., Lu, P.: On the approximability of budget feasible mechanisms. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pp. 685–699 (2011)Google Scholar
  10. 10.
    Cressie, N.A.: Statistics for Spatial Data. Wiley, Hoboken (1993)MATHGoogle Scholar
  11. 11.
    Dobzinski, S., Papadimitriou, C.H., Singer, Y.: Mechanisms for complement-free procurement. In: Proceedings 12th ACM Conference on Electronic Commerce (EC-2011), pp. 273–282 (2011)Google Scholar
  12. 12.
    Feige, U., Mirrokni, V.S., Vondrák, J.: Maximizing non-monotone submodular functions. SIAM J. Comput. 40(4), 1133–1153 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Feldman, M., Naor, J., Schwartz, R.: A unified continuous greedy algorithm for submodular maximization. In: IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pp. 570–579. IEEE Computer Society (2011)Google Scholar
  14. 14.
    Fujishige, S.: Canonical decompositions of symmetric submodular systems. Discret. Appl. Math. 5, 175–190 (1983)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Goel, G., Nikzad, A., Singla, A.: Mechanism design for crowdsourcing markets with heterogeneous tasks. In: Proceedings of the Second AAAI Conference on Human Computation and Crowdsourcing, HCOMP 2014 (2014)Google Scholar
  16. 16.
    Gupta, A., Nagarajan, V., Singla, S.: Adaptivity gaps for stochastic probing: submodular and XOS functions. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, pp. 1688–1702 (2017)Google Scholar
  17. 17.
    Gupta, A., Roth, A., Schoenebeck, G., Talwar, K.: Constrained non-monotone submodular maximization: offline and secretary algorithms. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 246–257. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-17572-5_20 CrossRefGoogle Scholar
  18. 18.
    Horel, T., Ioannidis, S., Muthukrishnan, S.: Budget feasible mechanisms for experimental design. In: Pardo, A., Viola, A. (eds.) LATIN 2014. LNCS, vol. 8392, pp. 719–730. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-54423-1_62 CrossRefGoogle Scholar
  19. 19.
    Jalaly, P., Tardos, E.: Simple and Efficient Budget Feasible Mechanisms for Monotone Submodular Valuations. arXiv:1703:10681 (2017)
  20. 20.
    Kleinberg, J., Tardos, E.: Algorithm Design. Addison Wesley, Boston (2006)Google Scholar
  21. 21.
    Krause, A., Singh, A.P., Guestrin, C.: Near-optimal sensor placements in Gaussian processes: theory, efficient algorithms and empirical studies. J. Mach. Learn. Res. 9, 235–284 (2008)MATHGoogle Scholar
  22. 22.
    Kulik, A., Shachnai, H., Tamir, T.: Approximations for monotone and nonmonotone submodular maximization with knapsack constraints. Math. Oper. Res. 38(4), 729–739 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lee, J., Mirrokni, V.S., Nagarajan, V., Sviridenko, M.: Maximizing nonmonotone submodular functions under matroid or knapsack constraints. SIAM J. Discret. Math. 23(4), 2053–2078 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Leonardi, S., Monaco, G., Sankowski, P., Zhang, Q.: Budget Feasible Mechanisms on Matroids. arXiv:1612:03150 (2016)
  25. 25.
    Myerson, R.: Optimal auction design. Math. Oper. Res. 6(1), 58–73 (1981)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions - I. Math. Program. 14(1), 265–294 (1978)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Queyranne, M.: Minimizing symmetric submodular functions. Math. Program. 82(1–2), 3–12 (1998)MathSciNetMATHGoogle Scholar
  28. 28.
    Schäffer, A.A., Yannakakis, M.: Simple local search problems that are hard to solve. SIAM J. Comput. 20(1), 56–87 (1991)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Singer, Y.: Budget feasible mechanisms. In: 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, pp. 765–774 (2010)Google Scholar
  30. 30.
    Singer, Y.: How to win friends and influence people, truthfully: influence maximization mechanisms for social networks. In: Proceedings of the 5th International Conference on Web Search and Web Data Mining, WSDM 2012, pp. 733–742 (2012)Google Scholar
  31. 31.
    Singla, A., Krause, A.: Incentives for privacy tradeoff in community sensing. In: Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing, HCOMP 2013. AAAI (2013)Google Scholar
  32. 32.
    Sviridenko, M.: A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett. 32(1), 41–43 (2004)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wolsey, L.A.: Maximising real-valued submodular functions: primal and dual heuristics for location problems. Math. Oper. Res. 7(3), 410–425 (1982)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Georgios Amanatidis
    • 1
  • Georgios Birmpas
    • 1
  • Evangelos Markakis
    • 1
  1. 1.Department of InformaticsAthens University of Economics and BusinessAthensGreece

Personalised recommendations