Abstract
We study the maximum budgeted allocation problem, i.e., the problem of selling a set of m indivisible goods to n players, each with a separate budget, such that we maximize the collected revenue. Since the natural assignment LP is known to have an integrality gap of \(\frac{3}{4}\), which matches the best known approximation algorithms, our main focus is to improve our understanding of the stronger configuration LP relaxation. In this direction, we prove that the integrality gap of the configuration LP is strictly better than \(\frac{3}{4}\), and provide corresponding polynomial time roundings, in the following restrictions of the problem: (i) the restricted budgeted allocation problem, in which all the players have the same budget and every item has the same value for any player it can be sold to, and (ii) the graph MBA problem, in which an item can be assigned to at most 2 players. Finally, we improve the best known upper bound on the integrality gap for the general case from \(\frac{5}{6}\) to \(2\sqrt{2}-2\approx 0.828\) and also prove hardness of approximation results for both cases.
Similar content being viewed by others
References
Andelman, N., Mansour, Y.: Auctions with budget constraints. In: SWAT, pp. 26–38 (2004)
Asadpour, A., Feige, U., Saberi, A.: Santa claus meets hypergraph matchings. In: APPROX-RANDOM, pp. 10–20. Springer (2008)
Azar, Y., Birnbaum, B.E., Karlin, A.R., Mathieu, C., Nguyen, C.T.: Improved approximation algorithms for budgeted allocations. ICALP 1, 186–197 (2008)
Bansal, N., Sviridenko, M.: The santa claus problem. In: Kleinberg, J.M. (ed.) STOC, pp. 31–40. ACM (2006)
Chakrabarty, D., Chuzhoy, J., Khanna, S.: On allocating goods to maximize fairness. In: 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25–27, 2009, Atlanta, Georgia, USA, pp. 107–116 (2009)
Chakrabarty, D., Goel, G.: On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and gap. SIAM J. Comput. 39(6), 2189–2211 (2010)
Ebenlendr, T., Krčál, M., Sgall, J.: Graph balancing: a special case of scheduling unrelated parallel machines. In: SODA, pp. 483–490. Society for Industrial and Applied Mathematics (2008)
Feige, U.: On allocations that maximize fairness. In: SODA, pp. 287–293. SIAM (2008)
Feige, U., Vondrák, J.: Approximation algorithms for allocation problems: improving the factor of 1–1/e. In: FOCS, pp. 667–676 (2006)
Gandhi, R., Khuller, S., Parthasarathy, S., Srinivasan, A.: Dependent rounding and its applications to approximation algorithms. J. ACM 53(3), 324–360 (2006)
Garg, R., Kumar, V., Pandit, V.: Approximation algorithms for budget-constrained auctions. In: APPROX-RANDOM, pp. 102–113. Springer (2001)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, Volume 2 of Algorithms and Combinatorics. Springer, Berlin (1993)
Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)
Lehmann, B., Lehmann, D.J., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. Games Econ. Behav. 55(2), 270–296 (2006)
Lenstra, J.K., Shmoys, D.B., Tardos, É.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46, 259–271 (1990)
Raz, R.: A parallel repetition theorem. SIAM J. Comput. 27(3), 763–803 (1998)
Shmoys, D.B., Tardos, É.: An approximation algorithm for the generalized assignment problem. Math. Program. 62, 461–474 (1993)
Srinivasan, A.: Budgeted allocations in the full-information setting. In: APPROX-RANDOM, pp. 247–253 (2008)
Svensson, O.: Santa claus schedules jobs on unrelated machines. SIAM J. Comput. 41(5), 1318–1341 (2012)
Trevisan, L.: Non-approximability results for optimization problems on bounded degree instances. In: STOC, pp. 453–461 (2001)
Trevisan, L., Sorkin, G.B., Sudan, M., Williamson, D.P.: Gadgets, approximation, and linear programming. SIAM J. Comput. 29(6), 2074–2097 (2000)
Verschae, J., Wiese, A.: On the configuration-LP for scheduling on unrelated machines. In: Algorithms-ESA, pp. 530–542. Springer (2011)
Acknowledgments
Many thanks go to Johan Håstad for finding an error in the proof of Theorem 4 in an earlier version of this paper. The authors would also like to thank the anonymous referees for many comments on improving the readability of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by ERC Advanced Investigator Grant 226203 and ERC Starting Grant 335288-OptApprox.
Rights and permissions
About this article
Cite this article
Kalaitzis, C., Ma̧dry, A., Newman, A. et al. On the configuration LP for maximum budgeted allocation. Math. Program. 154, 427–462 (2015). https://doi.org/10.1007/s10107-015-0928-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-015-0928-8