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On the configuration LP for maximum budgeted allocation

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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.

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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.

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Correspondence to Christos Kalaitzis.

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This research was partially supported by ERC Advanced Investigator Grant 226203 and ERC Starting Grant 335288-OptApprox.

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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

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  • DOI: https://doi.org/10.1007/s10107-015-0928-8

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