Membership Problems in Finite Groups

TL;DR

This paper establishes the computational complexity of various membership problems in finite permutation groups, showing NP-completeness and PSPACE-completeness results that deepen understanding of their algorithmic difficulty.

Contribution

It proves NP-completeness for subset sum, knapsack, and rational subset membership problems in permutation groups, and PSPACE-completeness for context-free membership, sharpening previous results.

Findings

Subset sum, knapsack, and rational subset problems are NP-complete.

Context-free membership problem is PSPACE-complete, NP-complete for certain classes.

New complexity bounds for DFA intersection and context-free grammar problems.

Abstract

We show that the subset sum problem, the knapsack problem and the rational subset membership problem for permutation groups are NP-complete. Concerning the knapsack problem we obtain NP-completeness for every fixed $n \geq 3$, where $n$ is the number of permutations in the knapsack equation. In other words: membership in products of three cyclic permutation groups is NP-complete. This sharpens a result of Luks, which states NP-completeness of the membership problem for products of three abelian permutation groups. We also consider the context-free membership problem in permutation groups and prove that it is PSPACE-complete but NP-complete for a restricted class of context-free grammars where acyclic derivation trees must have constant Horton-Strahler number. Our upper bounds hold for black box groups. The results for context-free membership problems in permutation groups yield new complexity bounds for various intersection non-emptiness problems for DFAs and a single context-free grammar.