Linear average-case complexity of algorithmic problems in groups

TL;DR

This paper investigates the average-case complexity of the word problem in various groups, demonstrating that it is often linear, and improves bounds on worst-case complexity for certain groups.

Contribution

It provides new insights into the average-case complexity of group problems, showing linearity in many cases and improving bounds for matrix groups.

Findings

Average-case complexity of the word problem is often linear in several group classes.

Improved bounds for worst-case complexity in groups of matrices, especially nilpotent groups.

Subgroup membership problem in free products also exhibits often linear average-case complexity.

Abstract

The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case time complexity of the word problem in several classes of groups and show that it is often the case that the average-case complexity is linear with respect to the length of an input word. The classes of groups that we consider include groups of matrices over rationals (in particular, polycyclic groups), some classes of solvable groups, as well as free products. Along the way, we improve several bounds for the worst-case complexity of the word problem in groups of matrices, in particular in nilpotent groups. For free products, we also address the average-case complexity of the subgroup membership problem and show that it is often linear, too. Finally, we discuss complexity of the identity problem that has not been considered before.