PLS-complete problems with lexicographic cost functions: Max-$k$-SAT and Abelian Permutation Orbit Minimization
Dominik Scheder, Johannes Tantow
TL;DR
The paper establishes strong hardness results for local search problems under lexicographic costs by building specialized reductions from FLIP-style problems to Max-k-SAT variants. It shows that Lexicographic 4-SAT/2-FLIP and 3-SAT/2-FLIP are PLS-complete, and that 4-SAT/1-FLIP is also PLS-complete, including tight and Abelian reductions. It further proves that Polm remains PLS-complete even when the permutations generating the orbit group are Abelian, including order-2, cyclic, and commuting cases. The authors reuse and adapt reduction techniques to reprove α-Nash equilibrium hardness in congestion games, extending the results to exponential costs, and thus connect lexicographic local-optimum problems to broader computational game-theoretic hardness. Overall, the work advances understanding of when lexicographic costs do or do not simplify local search and clarifies the impact of commutativity on Polm’s complexity, with implications for symmetry-breaking in SAT and related domains.
Abstract
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.