Abstract
We consider the Submodular Welfare Problem where we have $m$ items and $n$ players with given utility functions $w_i: 2^{[m]} \rightarrow \mathbb R_+$. The utility functions are assumed to be monotone and submodular. We want to find an allocation of disjoint sets $S_1, S_2, \ldots, S_n$ of items maximizing $\sum_i w_i(S_i)$. A $(1-1/\mathrm e )$-approximation for this problem in the demand oracle model has been given by Dobzinski and Schapira (2006). We improve this algorithm by presenting a $(1-1/\mathrm e + \epsilon)$-approximation for some small fixed $\epsilon>0$. We also show that the Submodular Welfare Problem is NP-hard to approximate within a ratio better than some $\rho < 1$. Moreover, this holds even when for each player there are only a constant number of items that have nonzero utility. The constant size restriction on utility functions makes it easy for players to efficiently answer any “reasonable” query about their utility functions. In contrast, for classes of instances that were used for previous hardness of approximation results, we present an incentive compatible (in expectation) mechanism based on fair division queries that achieves an optimal solution.
Highlights
We show that the Submodular Welfare Problem is NP -hard to approximate within a ratio better than some ρ < 1
In this paper we indicate how previous NP -hardness results can be modified to hold when utility functions are of constant size
Let us introduce a class of utility functions that is more general than submodular functions
Summary
It was not known whether this approximation factor can be improved in the demand oracle model It was not known whether the Configuration LP has an integrality gap arbitrarily close to 1 − 1/e for submodular utility functions. (A maximization problem is APX -hard if there is some constant ρ < 1 such that it is NP -hard to achieve approximation ratios better than ρ.) Contrasting Theorem 1.2 with the hardness results of [13] clearly shows that the best approximation ratio for Submodular Welfare depends on the query model that is allowed. Chakrabarty and Goel [3] proved that in the same model it is NP -hard to approximate the Submodular Welfare Problem within a ratio better than 15/16 In contrast to these results, we consider utility functions of a very restricted type.
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