Related papers: Quadratic equations over free groups are NP-comple…
We prove that the Diophantine problem for spherical quadratic equations in free metabelian groups is solvable and, moreover, NP-complete
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…
We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solutions sets of quadratic equations in a free group.
We prove that the quiver problem is NP complete.
We prove that in a torsion-free hyperbolic group $\Gamma$, the length of the value of each variable in a minimal solution of a quadratic equation $Q=1$ is bounded by $N|Q|^3$ for an orientable equation, and by $N|Q|^{4}$ for a…
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…
We study the quadratic residue problem known as an NP complete problem by way of the prime number and show that a nondeterministic polynomial process does not belong to the class P because of a random distribution of solutions for the…
We provide an algorithm which, for a given quadratic equation in the Grigorchuk group determines if it has a solution. As a corollary to our approach, we prove that the group has a finite commutator width.
Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of…
In this paper we study the conjugacy problem in polycyclic groups. Our main result is that we construct polycyclic groups $G_n$ whose conjugacy problem is at least as hard as the subset sum problem with $n$ indeterminates. As such, the…
We prove that, for every integer $n \ge 2$, a finite or infinite countable group $G$ can be embedded into a 2-generated group $H$ in such a way that the solvability of quadratic equations of length at most $n$ is preserved, i.e., every…
In this work we investigate tensor completions of groups by associative rings, which were introduced by R.Lyndon and G.Baumslag in 1960s. The main result states that there exists an algorithm that decides if a given finite system of…
We prove the decidability of the elementary theory of a free group.
Exponential equations in free groups were studied initially by Lyndon and Schutzenberger and then by Comerford and Edmunds. Comerford and Edmunds showed that the problem of determining whether or not the class of quadratic exponential…
NP complete problem is one of the most challenging issues. The question of whether all problems in NP are also in P is generally considered one of the most important open questions in mathematics and theoretical computer science as it has…
The square-free word problem relative to a system of two defining relations is decidable.
We consider pairs of finitely presented, residually finite groups $P\hookrightarrow\G$ for which the induced map of profinite completions $\hat P\to \hat\G$ is an isomorphism. We prove that there is no algorithm that, given an arbitrary…
We prove that persuasion is an NP-complete problem.
The Equation Problem in finitely presented groups asks if there exists an algorithm which determines in finite amount of time whether any given equation system has a solution or not. We show that the Equation Problem in central extensions…
Let $G$ be a non-trivial torsion free group and $t$ be an unknown. In this paper we consider three equations (over $G$) of arbitrary length and show that they have a solution (over $G$) provided two relations among their coefficients hold.…