可实现性带来不同：普遍服务义务中寻找汇点的复杂性差距
Realizability Makes a Difference: A Complexity Gap for Sink-Finding in USOs
摘要：
在超立方体的唯一汇点定向中寻找汇点的算法可以用来解决许多代数和几何问题，最重要的包括P-矩阵线性互补问题和线性规划。可实现的普遍服务义务是将这些问题归结为普遍服务义务的下沉问题而产生的义务。因此，找到可实现普遍服务义务的汇点具有高度的实际意义，但不知道是否可以通过算法利用可实现性来更快地找到汇点。然而，所有（非平凡的）已知的无条件下限下沉发现使用的USO是可证明的不可实现的。这表明，就可实现的普遍服务义务而言，寻找下沉点的问题可能确实比较容易。在本文中，我们表明，这是真实的所有USO的一个子类。我们考虑一类Matoušek型USO，这是一个翻译Matoušek的LP型问题到语言的USO。我们显示了一个查询复杂度之间的差距，所有的汇发现，和汇发现只有可实现的$n$维Matoušek型USO。我们提供了具体的确定性算法和下限的情况下，并表明，在可实现的情况下，$O(log^2 n)$顶点评估查询就足够了，而在一般情况下，正是$n$查询是必要的。Matoušek型普遍服务义务是发现的承认这种差距的第一个普遍服务义务类别。
英文摘要：
Algorithms for finding the sink in Unique Sink Orientations (USOs) of the hypercube can be used to solve many algebraic and geometric problems, most importantly including the P-Matrix Linear Complementarity Problem and Linear Programming. The realizable USOs are those that arise from the reductions of these problems to the USO sink-finding problem. Finding the sink of realizable USOs is thus highly practically relevant, yet it is unknown whether realizability can be exploited algorithmically to find the sink more quickly. However, all (non-trivial) known unconditional lower bounds for sink-finding make use of USOs that are provably not realizable. This indicates that the sink-finding problem might indeed be strictly easier on realizable USOs. In this paper we show that this is true for a subclass of all USOs. We consider the class of Matoušek-type USOs, which are a translation of Matoušek's LP-type problems into the language of USOs. We show a query complexity gap between sink-finding in all, and sink-finding in only the realizable $n$-dimensional Matoušek-type USOs. We provide concrete deterministic algorithms and lower bounds for both cases, and show that in the realizable case $O(log^2 n)$ vertex evaluation queries suffice, while in general exactly $n$ queries are needed. The Matoušek-type USOs are the first USO class found to admit such a gap.
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