How hard is it to find a local optimum? If we are given a graph and want to find a locally maximal cut--meaning that the number of edges in the cut can't be improved by moving a single vertex from one side to the other--then just iterating improving steps finds a local maximum in $ |E|$ steps. If, on the other hand, the edges are weighted, this problem becomes hard for the class PLS (Polynomial Local Search). We are interested in optimization problems with lexicographic costs. For Max-Cut this would mean that the edges $e_1,\dots, e_m$ have costs $c(e_i) = 2^i$. For such a cost function finding a global Max-Cut is easy. In contrast, we show that it is PLS-complete to find an assignment for a 4-CNF formula that is locally maximal (when the clauses have lexicographic weights); and also for a 3-CNF when we allow switching two variables at a time. We use these results to answer a question in Scheder and Tantow, who showed that finding a lexicographic local minimum of a string $s \in \{0,1\}^n$ under the action of a list of given permutations $π_1, \dots, π_k \in S_{n}$ is PLS-complete. They ask whether the problem stays PLS-complete when the $π_1,\dots,π_k$ commute, i.e., generate an Abelian subgroup $G$ of $S_n$. We show that it does, and in fact stays PLS-complete even (1) when every element in $G$ has order two or (2) when $G$ is cyclic. Additionally, we use it to further investigate the complexity of computing pure $α$-Nash equilibria in congestion games. Using lexicographic 4-SAT, we obtain a simple proof of the PLS-completeness originally shown by Skopalik and Vöcking that can be extended to exponential and polynomial delay functions with positive coefficients. The number of strategies per player and players per resource is bounded. However, the degree of the polynomials is not bounded by a constant.

翻译：寻找局部最优解有多困难？给定一个图，若希望找到局部最大的割——即无法通过将单个顶点从一侧移至另一侧来增加割边数量——那么仅通过重复改进步骤即可在$|E|$步内找到局部最大值。然而，若边带有权重，则此问题对PLS（多项式局部搜索）类而言变得困难。我们关注具有词序代价的优化问题。以最大割问题为例，这意味着边$e_1,\dots, e_m$的代价为$c(e_i) = 2^i$。对于此类代价函数，寻找全局最大割是容易的。相反，我们证明：对于词序加权子句的4-CNF公式，寻找局部最大真值赋值是PLS完全的；对于允许每次交换两个变量的3-CNF，该问题同样成立。我们利用这些结果回答了Scheder与Tantow提出的问题——他们曾证明，在给定置换列表$\pi_1, \dots, \pi_k \in S_{n}$作用下，寻找字符串$s \in \{0,1\}^n$的词序局部最小值是PLS完全的。他们追问：当$\pi_1,\dots,\pi_k$可交换（即生成$S_n$的阿贝尔子群$G$）时，问题是否仍保持PLS完全性？我们证明答案是肯定的，并且即使在以下两种情形下仍保持PLS完全：(1) $G$中每个元素阶为2，或(2) $G$为循环群。此外，我们利用该结果进一步研究拥塞博弈中计算纯$\alpha$-纳什均衡的复杂度。通过词序4-SAT，我们得到了Skopalik与Vöcking最初证明的PLS完全性的简洁证明，该方法可推广至具有正系数的指数延迟与多项式延迟函数。每个玩家的策略数和每个资源的玩家数均有界，但多项式的次数不限于常数。