arXiv subjects

Alexander Ushakov

Publications and source records attributed to Alexander Ushakov.

At least 19 recordsLinked to original sources

arXiv2026

One-variable equations over the lamplighter group

We study one-variable equations over the lamplighter group $L_2=\MZ_2 \wr \MZ$. While the decidability of arbitrary equations over $L_2$ remains open, we prove that the Diophantine problem for single equations in one variable is decidable. Our approach reduces the problem to a divisibility question for families of parametric Laurent polynomials over $\MZ_2$, whose exponents depend linearly on an integer parameter. To analyze this divisibility problem, we introduce a symbolic division procedure for associated bivariate polynomials and derive explicit bounds on the parameter from the structure of the resulting quotient and remainder. This yields an explicit decision procedure with exponential worst-case complexity. On the other hand, we show that for a generic class of equations, solvability can be decided in nearly quadratic time. These results establish a sharp contrast between worst-case and typical computational behavior and provide new tools for the study of equations over wreath products.

math.GR↗

arXiv2025

One variable equations over the lamplighter group

We prove that one variable equations in the lamplighter group $\MZ_2\wr \MZ$ are decidable and describe an algorithm for solving such equations. The algorithm has super-exponential time complexity in the worst case. We also show that, for most equations, decidability can be determined in nearly quadratic time; that is, the problem admits a nearly quadratic-time solution in the generic case.

math.GR↗

arXiv2025

HNN extensions of free groups with equal associated subgroups of finite index: polynomial time word problem

Let $G=F\ast_\varphi t$ be an HNN extension of a free group $F$ with two equal associated normal subgroups $H_1 = H_2$ of finite index. We prove that the word problem in $G$ is decidable in polynomial time. This result extends to the case where the subgroups $H_1=H_2$ are not normal, provided that the isomorphism $\varphi:H_1\to H_2$ satisfies an additional condition described in Section 5.

math.GR↗

arXiv2025

Orientable quadratic equations in wreath products

In this paper we study the complexity of solving orientable quadratic equations in wreath products $A\wr B$ of finitely generated abelian groups. We give a classification of cases (depending on genus and other characteristics of a given equation) when the problem is computationally hard or feasible.

math.GR↗

arXiv2024

Constrained inhomogeneous spherical equations: average-case hardness

In this paper we analyze computational properties of the Diophantine problem (and its search variant) for spherical equations $\prod_{i=1}^m z_i^{-1} c_i z_i = 1$ (and its variants) over the class of finite metabelian groups $G_{p,n}=\mathbb{Z}_p^n \rtimes \mathbb{Z}_p^\ast$, where $n\in\mathbb{N}$ and $p$ is prime. We prove that the problem of finding solutions for certain constrained spherical equations is computationally hard on average (assuming that some lattice approximation problem is hard in the worst case).

math.GR↗

arXiv2023

Quadratic equations in the lamplighter group

In this paper we study the complexity of solving quadratic equations in the lamplighter group. We give a complete classification of cases (depending on genus and other characteristics of a given equation) when the problem is $\mathbf{NP}$-complete or polynomial-time decidable. We notice that the conjugacy problem can be solved in linear time. Finally, we prove that the problem belongs to the class $\mathbf{XP}$.

math.GR↗

arXiv2023

Complexity of Spherical Equations in Finite Groups

In this paper we investigate computational properties of the Diophantine problem for spherical equations in some classes of finite groups. We classify the complexity of different variations of the problem, e.g., when $G$ is fixed and when $G$ is a part of the input. When the group $G$ is constant or given as multiplication table, we show that the problem always can be solved in polynomial time. On the other hand, for the permutation groups $S_n$ (with $n$ part of the input), the problem is NP-complete. The situation for matrix groups is quite involved: while we exhibit sequences of 2-by-2 matrices where the problem is NP-complete, in the full group $GL(2,p)$ ($p$ prime and part of the input) it can be solved in polynomial time. We also find a similar behaviour with subgroups of matrices of arbitrary dimension over a constant ring.

math.GR↗

arXiv2023

Quadratic equations in metabelian Baumslag-Solitar groups

For a finitely generated group $G$, the \emph{Diophantine problem} over $G$ is the algorithmic problem of deciding whether a given equation $W(z_1,z_2,\ldots,z_k) = 1$ (perhaps restricted to a fixed subclass of equations) has a solution in $G$. In this paper, we investigate the algorithmic complexity of the Diophantine problem for the class $\mathcal{C}$ of quadratic equations over the metabelian Baumslag-Solitar groups $\mathbf{BS}(1,n)$. We prove that this problem is $\mathbf{NP}$-complete whenever $n\neq \pm 1$, and determine the algorithmic complexity for various subclasses (orientable, nonorientable etc.) of $\mathcal{C}$.

math.GR↗

arXiv2022

The Diophantine problem for systems of algebraic equations with exponents

Consider the equation $q_1\alpha^{x_1}+\dots+q_k\alpha^{x_k} = q$, with constants $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$, $q_1,\ldots,q_k,q\in\overline{\mathbb{Q}}$ and unknowns $x_1,\ldots,x_k$, referred to in this paper as an \emph{algebraic equation with exponents}. We prove that the problem to decide if a given equation has an integer solution is $\textbf{NP}$-complete, and that the same holds for systems of equations (whether $\alpha$ is fixed or given as part of the input). Furthermore, we describe the set of all solutions for a given system of algebraic equations with exponents and prove that it is semilinear.

math.NT↗

arXiv2021

Linear time algorithm for the conjugacy problem in the first Grigorchuk group

We prove that the conjugacy problem in the first Grigorchuck group $Γ$ can be solved in linear time. Furthermore, the problem to decide if a list of elements $w_1,\ldots,w_k\inΓ$ contains a pair of conjugate elements can be solved in linear time. We also show that a conjugator for a pair of conjugate element $u,v\inΓ$ can be found in polynomial time.

math.GR↗

arXiv2020

On subset sum problem in branch groups

We consider a group-theoretic analogue of the classic subset sum problem. In this brief note, we show that the subset sum problem is NP-complete in the first Grigorchuk group. More generally, we show NP-hardness of that problem in weakly regular branch groups, which implies NP-completeness if the group is, in addition, contracting.

math.GR↗

arXiv2018

Orientable quadratic equations in free metabelian groups

We prove that the Diophantine problem for orientable quadratic equations in free metabelian groups is decidable and furthermore, NP-complete. In the case when the number of variables in the equation is bounded, the problem is decidable in polynomial time.

math.GR↗

arXiv2017

Subset sum problem in polycyclic groups

We consider a group-theoretic analogue of the classic subset sum problem. It is known that every virtually nilpotent group has polynomial time decidable subset sum problem. In this paper we use subgroup distortion to show that every polycyclic non-virtually-nilpotent group has NP-complete subset sum problem.

math.GR↗

arXiv2016

Conjugacy search problem and the Andrews-Curtis conjecture

We develop new computational methods for studying potential counterexamples to the Andrews-Curtis conjecture, in particular, Akbulut-Kurby examples AK(n). We devise a number of algorithms in an attempt to disprove the most interesting counterexample AK(3). To improve metric properties of the search space (which is a set of balanced presentations of the trivial group) we introduce a new transformation (called an ACM-move here) that generalizes the original Andrews-Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group. Finally, we prove that automorphism-moves can be applied to AK(n)-presentations. Unfortunately, despite a lot of effort we were unable to trivialize any of AK(n)-presentations, for n>2.

math.GR↗

arXiv2016

Generic case completeness

In this note we introduce a notion of a generically (strongly generically) NP-complete problem and show that the randomized bounded version of the halting problem is strongly generically NP-complete.

cs.CC↗

arXiv2015

Non-commutative lattice problems

We consider several subgroup-related algorithmic questions in groups, modeled after the classic computational lattice problems, and study their computational complexity. We find polynomial time solutions to problems like finding a subgroup element closest to a given group element, or finding a shortest non-trivial element of a subgroup in the case of nilpotent groups, and a large class of surface groups and Coxeter groups. We also provide polynomial time algorithm to compute geodesics in given generators of a subgroup of a free group.

math.GR↗

arXiv2015

Knapsack problems in products of groups

The classic knapsack and related problems have natural generalizations to arbitrary (non-commutative) groups, collectively called knapsack-type problems in groups. We study the effect of free and direct products on their time complexity. We show that free products in certain sense preserve time complexity of knapsack-type problems, while direct products may amplify it. Our methods allow to obtain complexity results for rational subset membership problem in amalgamated free products over finite subgroups.

math.GR↗

arXiv2015

Analysis of a certain polycyclic-group-based cryptosystem

We investigate security properties of the Anshel-Anshel-Goldfeld commutator key-establishment protocol used with certain polycyclic groups. We show that despite low success of the length based attack the protocol can be broken by a deterministic polynomial-time algorithm.

math.GR↗