Related papers: Quadratic equations over free groups are NP-comple…
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…
We give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity…
From the NP-hardness of the quantum separability problem and the relation between bipartite entanglement and the secret key correlations, it is shown that the problem deciding whether a given quantum state has secret correlations in it or…
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…
Friedl and L\"oh (2021, Confl. Math.) prove that testing whether or not there is an epimorphism from a finitely presented group to a virtually cyclic group, or to the direct product of an abelian and a finite group, is decidable. Here we…
We investigate the ability of a free pro-$\CC$ group of infinite rank to abstractly solve abstract embedding problems, and conclude that for some varieties $\CC$, the profinite completion of any order, of a free pro-$\CC$ group of infinite…
We prove that the compressed word problem and the compressed simultaneous conjugacy problem are solvable in polynomial time in hyperbolic groups. In such problems, group elements are input as words defined by straight line programs defined…
We introduce a new decision problem, called Packed Interval Covering (PIC) and show that it is NP-complete.
We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time…
The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsion-free groups is proved. In a special…
We show that the conjugacy problem is solvable in [finitely generated free]-by-cyclic groups, by using a result of O. Maslakova that one can algorithmically find generating sets for the fixed subgroups of free group automorphisms, and one…
We convert, within polynomial-time and sequential processing, NP-Complete Problems into a problem of deciding feasibility of a given system S of linear equations with constants and coefficients of binary-variables that are 0, 1, or -1. S is…
We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as Implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ…
This paper investigates the computational complexity of deciding whether the vertices of a graph can be partitioned into a disjoint union of cliques and a triangle-free subgraph. This problem is known to be $\NP$-complete on arbitrary…
Given a linear equation L, a set A of integers is L-free if A does not contain any non-trivial solutions to L. Meeks and Treglown showed that for certain kinds of linear equations, it is NP-complete to decide if a given set of integers…
We demonstrate a polynomial approach to express the decision version of the directed Hamiltonian Cycle Problem (HCP), which is NP-Complete, as the Solvability of a Polynomial Equation with a constant number of variables, within a bounded…
The Profinite Isomorphism Problem for a class of groups \mathcal{C} asks for an algorithm that decides for any two groups in \mathcal{C} whether they have isomorphic profinite completions. We present the positive solution to this problem…
0-1 Knapsack is a fundamental NP-complete problem. In this article we prove that it remains NP-complete even when the weights of the objects in the packing constraints and their values in the objective function satisfy specific stringent…
We investigate which free constructions (amalgamated products and HNN-extensions) over word hyperbolic groups produce groups that are again word hyperbolic. A complete answer is obtained for the case when the amalgamated subgroups are…
The problem whether a given permutation group contains a permutation with a given cycle type is studied. This problem is known to be NP-complete. In this paper it is shown that the problem can be solved in logspace for a cyclic permutation…