In this paper we consider the following maximum budgeted allocation<\/i> (MBA) problem: Given a set of \\(m\\) indivisible items and \\(n\\) agents, with each agent \\(i\\) willing to pay \\(b_{ij}\\) on item \\(j\\) and with a maximum budget of \\(B_i\\), the goal is to allocate items to agents to maximize revenue. The problem naturally arises as auctioneer revenue maximization in budget-constrained auctions and as the winner determination problem in combinatorial auctions when utilities of agents are budgeted-additive. Our main results are as follows: (i) We give a $3\/4$-approximation algorithm for MBA improving upon the previous best of \\(\\simeq0.632\\) [N. Andelman and Y. Mansour, Proceedings of the <\/i>9th Scandinavian Workshop on Algorithm Theory (SWAT)<\/i>, 2004, pp. 26-38], [J. Vondr\u00e1k, Proceedings of the <\/i>40th Annual ACM Symposium on the Theory of Computing (STOC)<\/i>, 2008, pp. 67-74] (also implied by the result of [U. Feige and J. Vondr\u00e1k, Proceedings of the <\/i>47th IEEE Symposium on Foundations of Computer Science (FOCS)<\/i>, 2006, pp. 667-676]). Our techniques are based on a natural LP relaxation of MBA, and our factor is optimal in the sense that it matches the integrality gap of the LP. (ii) We prove it is NP-hard to approximate MBA to any factor better than $15\/16$; previously only NP-hardness was known [T. Sandholm and S. Suri, Games Econom. Behav.<\/i>, 55 (2006), pp. 321-330], [B. Lehmann, D. Lehmann, and N. Nisan, Proceedings of the <\/i>3rd ACM Conference on Electronic Commerce (EC)<\/i>, 2001, pp. 18-28]. Our result also implies NP-hardness of approximating maximum submodular welfare with demand oracle<\/i> to a factor better than $15\/16$, improving upon the best known hardness of $275\/276$ [U. Feige and J. Vondr\u00e1k, Proceedings of the <\/i>47th IEEE Symposium on Foundations of Computer Science (FOCS)<\/i>, 2006, pp. 667-676]. (iii) Our hardness techniques can be modified to prove that it is NP-hard to approximate the generalized assignment problem<\/i> (GAP) to any factor better than $10\/11$. This improves upon the $422\/423$ hardness of [C. Chekuri and S. Khanna, Proceedings of the <\/i>11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)<\/i>, 2000, pp. 213-222], [M. Chleb\u00edk and J. Chleb\u00edkov\u00e1, Proceedings of the <\/i>8th Scandinavian Workshop on Algorithm Theory (SWAT)<\/i>, 2002, pp. 170-179]. We use iterative rounding<\/i> on a natural LP relaxation of the MBA problem to obtain the $3\/4$-approximation. We also give a \\((3\/4-\\epsilon)\\)-factor algorithm based on the primal-dual schema which runs in \\(\\tilde{O}(nm)\\) time, for any constant \\(\\epsilon>0\\).<\/p>\n","protected":false},"excerpt":{"rendered":"
In this paper we consider the following maximum budgeted allocation (MBA) problem: Given a set of indivisible items and agents, with each agent willing to pay on item and with a maximum budget of , the goal is to allocate items to agents to maximize revenue. 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