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arXiv 1702.06017cs.CC
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CLS：新问题与完整性

CLS: New Problems and Completeness

John Fearnley, Spencer Gordon, Ruta Mehta, Rahul Savani

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摘要：

复杂性类CLS是由Daskalakis和Papadimitriou引入的，目的是捕获PPAD $~\cap~$ PLS中一些众所周知的问题的复杂性，这些问题在某些情况下几十年来一直抵制将它们放在多项式时间中的尝试。没有完整的问题是已知的CLS，并在以前的工作中，问题ContractionMap，即，寻找压缩映射的近似不动点的问题，以及PLCP，即，解决P-矩阵线性互补问题的问题，被确定为主要候选人。首先，我们提出了一个新的CLS-完全问题MetaMetricContractionMap，这是密切相关的ContractionMap。其次，我们介绍EndOfPotentialLine，它通过单调有向路径直接捕获PPAD和PLS的各个方面，并表明EndOfPotentialLine通过双向减少EndOfMeteredLine在CLS中。后者被定义为通过限制势函数跟踪顶点在PPAD路径上的距离。第三，我们将PLCP减少到EndOfPotentialLine，从而使EndOfPotentialLine和EndOfMeteredLine至少与PLCP一样可能对CLS很难。最后一个结果利用了PLCP问题的Lemke路径的单调结构，使得EndOfPotentialLine成为捕获PLCP精确复杂性的可能候选者;我们注意到，在两人游戏中找到纳什均衡的Lemke-Howson路径的结构非常直接地激发了复杂性类PPAD的定义，最终准确地捕获了这个问题的复杂性。

英文摘要：

The complexity class CLS was introduced by Daskalakis and Papadimitriou with the goal of capturing the complexity of some well-known problems in PPAD$~\cap~$PLS that have resisted, in some cases for decades, attempts to put them in polynomial time. No complete problem was known for CLS, and in previous work, the problems ContractionMap, i.e., the problem of finding an approximate fixpoint of a contraction map, and PLCP, i.e., the problem of solving a P-matrix Linear Complementarity Problem, were identified as prime candidates. First, we present a new CLS-complete problem MetaMetricContractionMap, which is closely related to the ContractionMap. Second, we introduce EndOfPotentialLine, which captures aspects of PPAD and PLS directly via a monotonic directed path, and show that EndOfPotentialLine is in CLS via a two-way reduction to EndOfMeteredLine. The latter was defined to keep track of how far a vertex is on the PPAD path via a restricted potential function. Third, we reduce PLCP to EndOfPotentialLine, thus making EndOfPotentialLine and EndOfMeteredLine at least as likely to be hard for CLS as PLCP. This last result leverages the monotonic structure of Lemke paths for PLCP problems, making EndOfPotentialLine a likely candidate to capture the exact complexity of PLCP; we note that the structure of Lemke-Howson paths for finding a Nash equilibrium in a two-player game very directly motivated the definition of the complexity class PPAD, which eventually ended up capturing this problem's complexity exactly.

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