超越次模性/次可加性捕捉集函数的互补性
Capturing Complementarity in Set Functions by Going Beyond Submodularity/Subadditivity
摘要：
我们引入了两个新的“互补度”的措施，我们称之为，分别为超模宽度和超加宽度。两者都是根据互补性的自然证人制定的。我们证明了这两种措施是强大的证明，他们分别从次模和次加性的单调集函数的间隙的特征。因此，他们定义了单调集函数上的两个新层次，我们将其称为超模宽（SMW）层次和超加宽（SAW）层次，层次的0级分别精确地位于子模函数和子加函数上。我们提出了一个全面的比较分析的SMW层次结构和超模度（SD）的层次结构，由Feige和Izsak定义。我们证明了SMW层次结构是严格的表达比SD层次结构。我们表明，以前的结果近似保证福利和约束最大化，以及关于无政府代价价格（PoA）的简单拍卖可以扩展到没有任何损失的SD层次结构的SMW层次结构。我们还建立几乎匹配的信息理论下限。这些近似和硬度的结果相结合，说明SMW层次结构提供了一个准确的表征“近子模块”需要最大化近似。虽然SD和SMW层次结构支持简单拍卖的PoA上的非平凡的界限，我们表明，我们的SAW层次结构似乎捕捉到更多的内在属性需要实现简单拍卖的效率。到目前为止，SAW层次结构提供了最好的依赖性的单标拍卖的POA，几乎是竞争力的最大超图（MPH）层次的同时项目第一价格拍卖（SIA）。我们提供了几乎紧的下限的PoA拍卖相对于SAW层次结构。
英文摘要：
We introduce two new "degree of complementarity" measures, which we refer to, respectively, as supermodular width and superadditive width. Both are formulated based on natural witnesses of complementarity. We show that both measures are robust by proving that they, respectively, characterize the gap of monotone set functions from being submodular and subadditive. Thus, they define two new hierarchies over monotone set functions, which we will refer to as Supermodular Width (SMW) hierarchy and Superadditive Width (SAW) hierarchy, with level 0 of the hierarchies resting exactly on submodular and subadditive functions, respectively. We present a comprehensive comparative analysis of the SMW hierarchy and the Supermodular Degree (SD) hierarchy, defined by Feige and Izsak. We prove that the SMW hierarchy is strictly more expressive than the SD hierarchy. We show that previous results regarding approximation guarantees for welfare and constrained maximization as well as regarding the Price of Anarchy (PoA) of simple auctions can be extended without any loss from the SD hierarchy to the SMW hierarchy. We also establish almost matching information-theoretical lower bounds. The combination of these approximation and hardness results illustrate that the SMW hierarchy provides an accurate characterization of "near submodularity" needed for maximization approximation. While SD and SMW hierarchies support nontrivial bounds on the PoA of simple auctions, we show that our SAW hierarchy seems to capture more intrinsic properties needed to realize the efficiency of simple auctions. So far, the SAW hierarchy provides the best dependency for the PoA of Single-bid Auction, and is nearly as competitive as the Maximum over Positive Hypergraphs (MPH) hierarchy for Simultaneous Item First Price Auction (SIA). We provide almost tight lower bounds for the PoA of both auctions with respect to the SAW hierarchy.
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