Complexity of some algorithmic problems in groups: a survey
Vladimir Shpilrain
TL;DR
The survey examines average-case and generic-case time complexity for central group-theoretic problems, notably the word problem, across non-amenable and amenable groups. It highlights a foundational result: when a group contains a non-amenable quotient with favorable word-problem complexity, average-case performance often matches the lower bound of the underlying class, sometimes achieving linear time, with concrete corollaries for braid groups and Artin groups of extra-large type. For amenable groups, the landscape is more nuanced, yet linear average-case behavior appears in many polycyclic and nilpotent settings, and even stronger results (e.g., constant average-case) emerge for specific problems like primitivity detection via Whitehead-type analyses. The paper also outlines open directions and practical approaches, including exploiting forbidden-subword structures, and analyzes growth in matrix products to inform algorithmic design and hashing applications, underscoring the relevance to both theory and cryptography.
Abstract
In this survey, we address the worst-case, average-case, and generic-case time complexity of the word problem and some other algorithmic problems in several classes of groups and show that it is often the case that the average-case complexity of the word problem is linear with respect to the length of an input word, which is as good as it gets if one considers groups given by generators and defining relations. At the same time, there are other natural algorithmic problems, for instance, the geodesic (decision) problem or Whitehead's automorphism problem, where the average-case time complexity can be sublinear, even constant.