The Word Problem for Products of Symmetric Groups

TL;DR

This paper investigates the NP-complete word problem for products of symmetric groups and identifies specific subproblems that can be solved efficiently, expanding understanding of their computational complexity.

Contribution

The paper introduces efficient algorithms for three subproblems of WPPSG, including cases with consecutive sets and properties like the Consecutive Ones Property.

Findings

WPPSG is NP-complete in general

Efficient algorithms exist for subproblems with specific set properties

Subproblems are hierarchically related, with solutions for simpler cases aiding complex ones

Abstract

The word problem for products of symmetric groups (WPPSG) is a well-known NP-complete problem. An input instance of this problem consists of ``specification sets'' $X_1,\ldots,X_m \seq \{1,\ldots,n\}$ and a permutation $\tau$ on $\{1,\ldots,n\}$. The sets $X_1,\ldots,X_m$ specify a subset of the symmetric group $\cS_n$ and the question is whether the given permutation $\tau$ is a member of this subset. We discuss three subproblems of WPPSG and show that they can be solved efficiently. The subproblem WPPSG$_0$ is the restriction of WPPSG to specification sets all of which are sets of consecutive integers. The subproblem WPPSG$_1$ is the restriction of WPPSG to specification sets which have the Consecutive Ones Property. The subproblem WPPSG$_2$ is the restriction of WPPSG to specification sets which have what we call the Weak Consecutive Ones Property. WPPSG$_1$ is more general than WPPSG$_0$ and WPPSG$_2$ is more general than WPPSG$_1$. But the efficient algorithms that we use for solving WPPSG$_1$ and WPPSG$_2$ have, as a sub-routine, the efficient algorithm for solving WPPSG$_0$.