PLS-Completeness of String Permutations

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.