电位线终点
End of Potential Line
摘要：
我们介绍了EndOfPotentialLine问题和相应的复杂性类EOPL的所有问题，可以减少到它在多项式时间。这个类捕获的问题，承认一个单一的组合证明他们的联合成员资格的复杂性类PPAD的不动点问题和PLS的局部搜索问题。EOPL是一个组合定义的替代类CLS（连续局部搜索），这是引入的目标，捕获的复杂性，一些众所周知的问题，在PPAD $\cap$ PLS抵制，在某些情况下，几十年来，试图把他们在多项式时间。其中两个是收缩，找到收缩映射的不动点的问题，以及P-LCP，解决P-矩阵线性互补问题的问题。我们表明，EndOfPotentialLine是在CLS通过一个双向减少EndOfMeteredLine。后者是在中定义的，用于显示CLS的查询和加密下限。我们的两个主要结果表明，PL-收缩（分段线性收缩）和P-LCP都在EOPL中。我们的减少意味着PL-收缩和P-LCP的承诺版本在承诺类UniqueEOPL中，这对应于单个潜在线的情况。这也表明，简单随机，贴现，平均支付，平价游戏是EOPL。使用我们的减少PL-收缩的见解，我们得到了第一个多项式时间算法，用于在固定的尺寸为任何$\ell_p$规范，在以前这样的算法只知道的$\ell_2$和$\ell_\infty$规范的收缩映射的不动点。我们从P-LCP到EndOfPotentialLine的简化允许应用Aldous技术，这反过来又给出了P-LCP的最快已知随机算法。
英文摘要：
We introduce the problem EndOfPotentialLine and the corresponding complexity class EOPL of all problems that can be reduced to it in polynomial time. This class captures problems that admit a single combinatorial proof of their joint membership in the complexity classes PPAD of fixpoint problems and PLS of local search problems. EOPL is a combinatorially-defined alternative to the class CLS (for Continuous Local Search), which was introduced in 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. Two of these are Contraction, the problem of finding a fixpoint of a contraction map, and P-LCP, the problem of solving a P-matrix Linear Complementarity Problem. We show that EndOfPotentialLine is in CLS via a two-way reduction to EndOfMeteredLine. The latter was defined in to show query and cryptographic lower bounds for CLS. Our two main results are to show that both PL-Contraction (Piecewise-Linear Contraction) and P-LCP are in EOPL. Our reductions imply that the promise versions of PL-Contraction and P-LCP are in the promise class UniqueEOPL, which corresponds to the case of a single potential line. This also shows that simple-stochastic, discounted, mean-payoff, and parity games are in EOPL. Using the insights from our reduction for PL-Contraction, we obtain the first polynomial-time algorithms for finding fixed points of contraction maps in fixed dimension for any $\ell_p$ norm, where previously such algorithms were only known for the $\ell_2$ and $\ell_\infty$ norms. Our reduction from P-LCP to EndOfPotentialLine allows a technique of Aldous to be applied, which in turn gives the fastest-known randomized algorithm for the P-LCP.
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