SIAM Journal on Computing, Ahead of Print.
We study a central problem in algorithmic mechanism design: constructing truthful mechanisms for welfare maximization in combinatorial auctions with submodular bidders. Dobzinski, Nisan, and Schapira provided the first mechanism that guarantees a nontrivial approximation ratio of $O(\log^2 m)$ [STOC'06, ACM, New York, 2006, pp. 644--652], where $m$ is the number of items. This approximation ratio was subsequently improved to $O(\log m\log \log m)$ [S. Dobzinski, APPROX'07, Springer, Berlin, 2007, pp. 89--103] and then to $O(\log m)$ [P. Krysta and B. Vöcking, ICALP'12, Springer, Heidelberg, 2012, pp. 636--647]. In this paper we develop the first mechanism that breaks the logarithmic barrier. Specifically, the mechanism provides an approximation ratio of $O(\sqrt {\log m})$. Similarly to previous constructions, our mechanism uses polynomially many value and demand queries and, in fact, provides the same approximation ratio for the larger class of XOS (also known as fractionally subadditive) valuations. We also develop a computationally efficient implementation of the mechanism for combinatorial auctions with budget additive bidders. Although, in general, computing a demand query is NP-hard for budget additive valuations, we observe that the specific form of demand queries that our mechanism uses can be efficiently computed when bidders are budget additive.

中文翻译：

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借助亚模块竞标者打破真实组合拍卖的对数壁垒

《 SIAM计算杂志》，预印本。
我们研究算法机制设计中的一个核心问题：在具有次模块竞标者的组合拍卖中构建使福利最大化的真实机制。Dobzinski，Nisan和Schapira提供了第一种机制，可确保$ O（\ log ^ 2 m）$的非平凡逼近比[STOC'06，ACM，纽约，2006年，第644--652页]，其中$ m $是项目数。随后将该近似比提高到$ O（\ log m \ log \ log m）$ [S。Dobzinski，APPROX'07，施普林格，柏林，2007年，第89--103页]，然后是$ O（\ log m）$ [P. Krysta和B.Vöcking，《 ICALP'12》，施普林格，海德堡，2012年，第636--647页]。在本文中，我们开发了第一个打破对数障碍的机制。具体地说，该机制提供了$ O（\ sqrt {\ log m}）$的近似比率。与以前的结构类似，我们的机制使用多项式许多值和需求查询，并且实际上为较大的XOS（也称为分数次加法）估值提供了相同的近似值。我们还开发了一种具有预算增效竞标者的组合拍卖机制的高效计算实现。尽管通常来说，对于预算添加剂估值，计算需求查询是NP难的，但是我们观察到，当投标人是预算添加剂时，我们的机制使用的特定形式的需求查询可以有效地计算出来。