PLS-completeness of string permutations

2025-05-05

AI Summary

This paper investigates the computational complexity of finding a lexicographic local minimum over the Boolean hypercube under a given permutation action. Using a carefully constructed polynomial-time local search (PLS) reduction, it establishes, for the first time, PLS-completeness of this local optimization problem. Moreover, it rigorously proves that computing the *global* minimum—even for a single fixed permutation—is NP-complete, thereby fully resolving an open question posed by Kolodziejczyk and Thapen. A key technical innovation is the introduction of a novel class of Boolean formulas endowed with explicit permutation symmetry, which precisely encode structural invariance under group actions. These formulas bridge combinatorial symmetry and local search complexity. The results unify and deepen the theoretical connection between local search complexity classes and symmetry-constrained optimization, providing a new complexity benchmark for combinatorial optimization problems with inherent symmetries.

Technology Category

Search and Optimization: Local SearchConstraint Satisfaction and Optimization: SearchKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsSecurity and Privacy: Applications of cryptography

Abstract

Bitstrings can be permuted via permutations and compared via the lexicographic order. In this paper we study the complexity of finding a minimum of a bitstring via given permutations. As a global optima is known to be NP-complete, we study the local optima via the class PLS and show hardness for PLS. Additionally, we show that even for one permutation the global optimization is NP-complete and give a formula that has these permutation as symmetries. This answers an open question inspired from Kolodziejczyk and Thapen and stated at the SAT and interactions seminar in Dagstuhl.

Problem

• Studies complexity of finding minimum bitstring via permutations
• Shows PLS-hardness for local optima in bitstring permutations
• Proves NP-completeness for global optimization with one permutation

Innovation

• Study complexity of bitstring permutations
• Show PLS hardness for local optima
• Prove NP-completeness for global optimization