PLS-complete problems with lexicographic cost functions: Max-$k$-SAT and Abelian Permutation Orbit Minimization

2025-10-17

AI Summary

This paper investigates PLS-completeness under lexicographic cost functions, focusing on two fundamental settings: (1) finding a locally optimal truth assignment for 4-CNF formulas with lexicographic weights, and (2) computing the lexicographically smallest string within an orbit induced by an abelian or cyclic group action. Via polynomial-time PLS reductions, we establish that Lexicographically Weighted Max-4-SAT—and even Max-3-SAT under relaxed neighborhood definitions—is PLS-complete. Crucially, we resolve a long-standing open problem by proving that lexicographic orbit minimization under commutative permutation groups (including cyclic groups) remains PLS-complete. These results systematically characterize the local search complexity of combinatorial optimization problems under lexicographic costs, significantly extending the theoretical understanding of the PLS class in settings involving group actions and weighted Boolean satisfiability.

Technology Category

Search and Optimization: Local SearchConstraint Satisfaction and Optimization: SatisfiabilityPlanning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

Economics, Online Markets and Human Computation: Cost models of using LLMs in production systemsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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)[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 $π_1, dots, π_k in S_{n}$ 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 permutations $π$.

Problem

• Proving PLS-completeness for lexicographic Max-4-SAT local optimization
• Establishing PLS-completeness for Abelian permutation orbit minimization
• Analyzing computational complexity of local search with lexicographic costs

Innovation

• Proving PLS-completeness for lexicographic Max-4-SAT
• Extending PLS-completeness to Abelian permutation groups
• Analyzing local search complexity under lexicographic weights