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)[16]. 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 $c(e_i) = 2^{m-i}$. For such a cost function, it is easy to see that 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 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[15], 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 $\pi_1, \dots, \pi_k \in S_{n}$ is PLS-complete. They ask whether the problem stays PLS-complete when the $\pi_1,\dots,\pi_k$ 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 $\pi_1,\dots,\pi_k$ are powers of a single permutations $\pi$.

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改善割中的边数——则仅需迭代改进步骤即可找到局部最大值，因为割的规模至多增加$|E|$次。然而，若边被赋予权重，该问题对PLS（多项式局部搜索）类而言将变得困难[16]。我们关注具有词典序代价的优化问题。对最大割而言，这意味着边$e_1,\dots, e_m$具有代价$c(e_i) = 2^{m-i}$。对于此类代价函数，容易看出寻找全局最大割是简单的。与之相对，我们证明：对于4-CNF公式（当子句具有词典序权重时）寻找局部最大赋值是PLS完全的；对于3-CNF公式，若放宽局部性定义（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[15]中的一个问题，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$s \in \{0,1\}^n$的词典序局部最小值是PLS完全的。他们询问当$\pi_1,\dots,\pi_k$可交换（即生成$S_n$的一个阿贝尔子群$G$）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，且甚至在下述两种情形下依然成立：（1）当$G$中每个元素的阶均为二；（2）当$G$是循环群，即所有$\pi_1,\dots,\pi_k$均为单个置换$\pi$的幂次。