PLS-completeness of string permutations

TL;DR

This paper investigates the computational complexity of optimizing bitstring permutations, demonstrating NP-completeness and PLS-hardness for global and local optima, respectively, and explores symmetry properties in permutation-based optimization.

Contribution

It establishes the PLS-completeness of local optimization for bitstring permutations and proves NP-completeness for global optimization even with a single permutation, answering an open problem.

Findings

Global optimization is NP-complete for bitstring permutations.

Local optima finding is PLS-hard, indicating computational difficulty.

Even with one permutation, the problem remains NP-complete.

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.