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 cannot be improved by moving a single vertex from one side to the other--then just iterating improving steps finds a local maximum since the size of the cut can increase at most $|E|$ times. 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 {\em lexicographic costs}. For Max-Cut this would mean that the edges $e_1,\dots, e_m$ have costs $c(e_i) = 2^{m-i}$. For such a cost function, it is easy to see that finding a {\em 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 relax the notion of ``local'' by allowing to switch 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 $S_n$. In this work, we show that it does, and in fact stays PLS-complete even (1) when every element in $G$ has order two and also (2) when $G$ is cyclic, i.e., all $π_1,\dots,π_k$ are powers of a single permutation $π$. Additionally, we use it to reprove the hardness of computing an $α$ approximate Nash equilibria in congestion games by Skopalik and Vöcking and extend the result from step functions to exponential functions.

翻译：寻找局部最优解的难度如何？若给定一个图并希望找到一个局部最大割——即无法通过将单个顶点从一侧移至另一侧来提升割边数量——则仅需迭代改进步骤即可找到局部最大值，因为割的规模最多只能增加$|E|$次。然而，若边被赋予权重，该问题即成为PLS（多项式局部搜索）类中的困难问题。我们关注具有{\\em 字典序成本}的优化问题。对于最大割问题，这意味着边$e_1,\\dots, e_m$具有成本$c(e_i) = 2^{m-i}$。对此类成本函数，易证寻找{\\em 全局}最大割是简单的。相比之下，我们证明：对于4-CNF公式（当子句具有字典序权重时）寻找局部最大赋值是PLS完全的；对于3-CNF公式，若放宽“局部”定义允许同时切换两个变量，该问题同样PLS完全。我们利用这些结果回答了Scheder与Tantow提出的问题：他们证明了在给定置换列表$\\pi_1, \\dots, \\pi_k \\in S_{n}$作用下，寻找字符串$s \\in \\{0,1\\}^n$的字典序局部最小值是PLS完全的。他们询问当$\\pi_1,\\dots,\\pi_k$可交换（即生成$S_n$的阿贝尔子群$G$）时，该问题是否仍保持PLS完全性。本工作中，我们证明其确实保持PLS完全性，且甚至在以下两种情况下依然成立：(1) $G$中每个元素的阶均为二；(2) $G$是循环群，即所有$\\pi_1,\\dots,\\pi_k$均为单个置换$\\pi$的幂。此外，我们借此重新证明了Skopalik与Vöcking关于计算拥塞博弈中$\\alpha$近似纳什均衡的困难性，并将结果从阶梯函数推广至指数函数。