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Unformatted text preview: Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders Shahar Dobzinski * Noam Nisan † Michael Schapira ‡ ABSTRACT We exhibit three approximation algorithms for the alloca- tion problem in combinatorial auctions with complement free bidders. The running time of these algorithms is polyno- mial in the number of items m and in the number of bidders n , even though the “input size” is exponential in m . The first algorithm provides an O (log m ) approximation. The second algorithm provides an O ( √ m ) approximation in the weaker model of value oracles. This algorithm is also incen- tive compatible. The third algorithm provides an improved 2-approximation for the more restricted case of “ XOS bid- ders”, a class which strictly contains submodular bidders. We also prove lower bounds on the possible approximations achievable for these classes of bidders. These bounds are not tight and we leave the gaps as open problems. Categories and Subject Descriptors F.2.8 [ Analysis of Algorithms and Problem complex- ity ]: Miscellaneous General Terms Algorithms Keywords Combinatorial Auctions * The School of Computer Science and Engineering, The Hebrew University of Jerusalem, [email protected] . Supported by grants from the Israel Science Foundation and the USA-Israel Bi-national Science Foundation. † The School of Computer Science and Engineering The He- brew University of Jerusalem, [email protected] . Sup- ported by grants from the Israel Science Foundation and the USA-Israel Bi-national Science Foundation. ‡ The School of Computer Science and Engineering The Hebrew University of Jerusalem [email protected] . Supported by grants from the Israel Science Foundation and the USA-Israel Bi-national Science Foundation. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. STOC’05, May 22-24, 2005, Baltimore, Maryland, USA. Copyright 2005 ACM 1-58113-960-8/05/0005 ... $ 5.00. 1. INTRODUCTION In a combinatorial auction, a set M of items, | M | = m , is sold to n bidders. The combinatorial character of the auc- tion comes from the fact that each bidder values bundles of items, rather than valuing items directly. I.e., the i ’th bid- der’s value for each bundle is given by a valuation function v i , where for each subset S ⊆ M , v i ( S ) denotes the value (maximum willingness to pay) of the bundle S for bidder i ....
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This note was uploaded on 10/25/2009 for the course EE ee taught by Professor Ericlu during the Spring '09 term at École Normale Supérieure.
- Spring '09