On the Approximability of Budgeted Allocations and Improved Lower Bounds for Submodular Welfare Maximization and GAP
Related Databases
Web of Science
You must be logged in with an active subscription to view this.Article Data
History
Publication Data
In this paper we consider the following maximum budgeted allocation (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 9th Scandinavian Workshop on Algorithm Theory (SWAT), 2004, pp. 26–38], [J. Vondrák, Proceedings of the 40th Annual ACM Symposium on the Theory of Computing (STOC), 2008, pp. 67–74] (also implied by the result of [U. Feige and J. Vondrák, Proceedings of the 47th IEEE Symposium on Foundations of Computer Science (FOCS), 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., 55 (2006), pp. 321–330], [B. Lehmann, D. Lehmann, and N. Nisan, Proceedings of the 3rd ACM Conference on Electronic Commerce (EC), 2001, pp. 18–28]. Our result also implies NP-hardness of approximating maximum submodular welfare with demand oracle to a factor better than $15/16$, improving upon the best known hardness of $275/276$ [U. Feige and J. Vondrák, Proceedings of the 47th IEEE Symposium on Foundations of Computer Science (FOCS), 2006, pp. 667–676]. (iii) Our hardness techniques can be modified to prove that it is NP-hard to approximate the generalized assignment problem (GAP) to any factor better than $10/11$. This improves upon the $422/423$ hardness of [C. Chekuri and S. Khanna, Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2000, pp. 213–222], [M. Chlebík and J. Chlebíková, Proceedings of the 8th Scandinavian Workshop on Algorithm Theory (SWAT), 2002, pp. 170–179]. We use iterative rounding 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 $\epsilon>0$.
Sign in
Help
View Cart
