Budget Feasible Mechanisms on Matroids

  • Stefano Leonardi
  • Gianpiero Monaco
  • Piotr Sankowski
  • Qiang ZhangEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)


Motivated by many practical applications, in this paper we study budget feasible mechanisms where the goal is to procure independent sets from matroids. More specifically, we are given a matroid \({\mathcal {M}}=(E,{\mathcal {I}})\) where each ground (indivisible) element is a selfish agent. The cost of each element (i.e., for selling the item or performing a service) is only known to the element itself. There is a buyer with a budget having additive valuations over the set of elements E. The goal is to design an incentive compatible (truthful) budget feasible mechanism which procures an independent set of the matroid under the given budget that yields the largest value possible to the buyer. Our result is a deterministic, polynomial-time, individually rational, truthful and budget feasible mechanism with 4-approximation to the optimal independent set. Then, we extend our mechanism to the setting of matroid intersections in which the goal is to procure common independent sets from multiple matroids. We show that, given a polynomial time deterministic blackbox that returns \(\alpha \)-approximation solutions to the matroid intersection problem, there exists a deterministic, polynomial time, individually rational, truthful and budget feasible mechanism with \((3\alpha +1)\)-approximation to the optimal common independent set.


  1. 1.
    Anari, N., Goel, G., Nikzad, A.: Mechanism design for crowdsourcing: an optimal 1-1/e competitive budget-feasible mechanism for large markets. In: 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 266–275. IEEE (2014)Google Scholar
  2. 2.
    Ausubel, L.M.: An efficient ascending-bid auction for multiple objects. Am. Econ. Rev. 94, 1452–1475 (2004)CrossRefGoogle Scholar
  3. 3.
    Bei, X., Chen, N., Gravin, N., Lu, P.: Budget feasible mechanism design: from prior-free to bayesian. In: Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing (STOC), pp. 449–458. ACM (2012)Google Scholar
  4. 4.
    Bikhchandani, S., de Vries, S., Schummer, J., Vohra, R.V.: An ascending vickrey auction for selling bases of a matroid. Oper. Res. 59(2), 400–413 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chan, H., Chen, J.: Truthful multi-unit procurements with budgets. In: Liu, T.-Y., Qi, Q., Ye, Y. (eds.) WINE 2014. LNCS, vol. 8877, pp. 89–105. Springer, Cham (2014). doi: 10.1007/978-3-319-13129-0_7 Google Scholar
  6. 6.
    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), pp. 685–699. SIAM (2011)Google Scholar
  7. 7.
    Clarke, E.H.: Multipart pricing of public goods. Public choice 11(1), 17–33 (1971)CrossRefGoogle Scholar
  8. 8.
    Demange, G., Gale, D., Sotomayor, M.: Multi-item auctions. J. Polit. Econ. 94, 863–872 (1986)CrossRefGoogle Scholar
  9. 9.
    Goel, G., Mirrokni, V., Leme, R.P.: Polyhedral clinching auctions and the adwords polytope. J. ACM (JACM) 62(3), 18 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Goel, G., Nikzad, A., Singla, A.: Allocating tasks to workers with matching constraints: truthful mechanisms for crowdsourcing markets. In: Proceedings of the Companion Publication of the 23rd International Conference on World Wide Web Companion, pp. 279–280 (2014)Google Scholar
  11. 11.
    Groves, T.: Incentives in teams. Econometrica: J. Econometric Soc. 41, 617–631 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Karlin, A.R., Kempe, D.: Beyond VCG: frugality of truthful mechanisms. In: 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 615–624. IEEE (2005)Google Scholar
  13. 13.
    Kleinberg, R., Weinberg, S.M.: Matroid prophet inequalities. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, pp. 123–136. ACM (2012)Google Scholar
  14. 14.
    Krysta, P., Zhang, J.: House markets with matroid and knapsack constraints. In: Proceedings of The 43rd International Colloquium on Automata, Languages and Programming (ICALP) (2016)Google Scholar
  15. 15.
    Schrijver, A.: Combinatorial Optimization. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)zbMATHGoogle Scholar
  16. 16.
    Singer, Y.: Budget feasible mechanisms. In: 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 765–774 (2010)Google Scholar
  17. 17.
    Tse, D.N.C., Hanly, S.V.: Multiaccess fading channels. I. Polymatroid structure, optimal resource allocation and throughput capacities. IEEE Trans. Inf. Theor. 44(7), 2796–2815 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Vickrey, W.: Counterspeculation, auctions, and competitive sealed tenders. J. Financ. 16(1), 8–37 (1961)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Stefano Leonardi
    • 1
  • Gianpiero Monaco
    • 2
  • Piotr Sankowski
    • 3
  • Qiang Zhang
    • 1
    Email author
  1. 1.Sapienza University of RomeRomeItaly
  2. 2.University of L’AquilaL’AquilaItaly
  3. 3.University of WarsawWarsawPoland

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