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Linear average-case complexity of algorithmic problems in groups

Alexander Olshanskii, Vladimir Shpilrain

TL;DR

This work systematically analyzes average-case time complexity for core algorithmic problems in groups, focusing on the word problem, subgroup membership, and the identity problem. It establishes that the average-case word problem is linear in many natural classes, including polycyclic, nilpotent, and free-product constructions, and it sharpens worst-case bounds for matrix groups using divide-and-conquer and fast-arithmetic techniques. The authors also extend linear-average-case results to Thompson's group $F$, discuss the subgroup-membership problem in free products, and develop density- and encoding-based methods to treat the identity problem across varieties, with several open questions framing future work. Collectively, the results advance understanding of practical algorithmic performance in computational group theory and provide techniques applicable across linear and solvable group settings.

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.

Linear average-case complexity of algorithmic problems in groups

TL;DR

This work systematically analyzes average-case time complexity for core algorithmic problems in groups, focusing on the word problem, subgroup membership, and the identity problem. It establishes that the average-case word problem is linear in many natural classes, including polycyclic, nilpotent, and free-product constructions, and it sharpens worst-case bounds for matrix groups using divide-and-conquer and fast-arithmetic techniques. The authors also extend linear-average-case results to Thompson's group , discuss the subgroup-membership problem in free products, and develop density- and encoding-based methods to treat the identity problem across varieties, with several open questions framing future work. Collectively, the results advance understanding of practical algorithmic performance in computational group theory and provide techniques applicable across linear and solvable group settings.

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.
Paper Structure (13 sections, 25 theorems, 15 equations)

This paper contains 13 sections, 25 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. The average-case complexity vs. the worst-case complexity
  3. The word problem in linear groups
  4. The word problem in nilpotent groups and in groups of unitriangular matrices
  5. The word problem in groups of matrices over integers or rationals
  6. The average-case complexity of the word problem in polycyclic groups
  7. The word problem in groups with a solvable factor group
  8. The word problem in groups that map onto a (generalized) lamplighter group
  9. The word problem in Thompson's group $F$
  10. The word problem in free products of groups
  11. The subgroup membership problem in free products of groups
  12. The identity problem
  13. Open problems

Key Result

Lemma 1

Let ${\bf B}=\{B_1, \ldots, B_s\}$ be a finite collection of uni(upper)triangular matrices. The absolute value of the $(i,j)$th entry in any product of $n$ matrices, each of which comes from this collection, is bounded by a polynomial in $n$ of degree $j-i$, and therefore the total bit length of the

Theorems & Definitions (46)

  • Lemma 1
  • proof
  • Lemma 2
  • Proposition 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Theorem 2
  • ...and 36 more