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2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06) (2006)
Berkeley, California
Oct. 21, 2006 to Oct. 24, 2006
ISSN: 0272-5428
ISBN: 0-7695-2720-5
pp: 667-676
Uriel Feige , Microsoft Research, USA
Jan Vondrak , Princeton University, USA
Combinatorial allocation problems require allocating items to players in a way that maximizes the total utility. Two such problems received attention recently, and were addressed using the same linear programming (LP) relaxation. In the Maximum Submodular Welfare (SMW) problem, utility functions of players are submodular, and for this case Dobzinski and Schapira [SODA 2006] showed an approximation ratio of 1 - 1/e. In the Generalized Assignment Problem (GAP) utility functions are linear but players also have capacity constraints. GAP admits a (1 - 1/e)- approximation as well, as shown by Fleischer, Goemans, Mirrokni and Sviridenko [SODA 2006]. In both cases, the approximation ratio was in fact shown for a more general version of the problem, for which improving 1 - 1/e is NPhard. <p>In this paper, we show how to improve the 1 - 1/e approximation ratio, both for SMW and for GAP. A common theme in both improvements is the use of a new and optimal Fair Contention Resolution technique. However, each of the improvements involves a different rounding procedure for the above mentioned LP.</p> <p>In addition, we prove APX-hardness results for SMW (such results were known for GAP). An important feature of our hardness results is that they apply even in very restricted settings, e.g. when every player has nonzero utility only for a constant number of items.</p>

U. Feige and J. Vondrak, "Approximation algorithms for allocation problems: Improving the factor of 1 - 1/e," 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)(FOCS), Berkeley, California, 2006, pp. 667-676.
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