The complexity of the equation solvability and equivalence problems over finite groups

2019 ◽  
Vol 30 (03) ◽  
pp. 607-623
Author(s):  
Attila Földvári ◽  
Gábor Horváth
Keyword(s):  
Abelian Group ◽  
Polynomial Time ◽  
Finite Groups ◽  
Finite Abelian Group ◽  
Equivalence Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Semidirect Products ◽  
Solvability Problem ◽  
Equation Solvability

We provide a polynomial time algorithm for deciding the equation solvability problem over finite groups that are semidirect products of a [Formula: see text]-group and an Abelian group. As a consequence, we obtain a polynomial time algorithm for deciding the equivalence problem over semidirect products of a finite nilpotent group and a finite Abelian group. The key ingredient of the proof is to represent group expressions using a special polycyclic presentation of these finite solvable groups.

scholarly journals The complexity of the equation solvability problem over semipattern groups

2017 ◽  
Vol 27 (02) ◽  
pp. 259-272 ◽  
Author(s):  
Attila Földvári
Keyword(s):  
Original Problem ◽  
Main Idea ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Matrix Groups ◽  
Semidirect Products ◽  
Solvability Problem ◽  
Equation Solvability

The complexity of the equation solvability problem is known for nilpotent groups, for not solvable groups and for some semidirect products of Abelian groups. We provide a new polynomial time algorithm for deciding the equation solvability problem over certain semidirect products, where the first factor is not necessarily Abelian. Our main idea is to represent such groups as matrix groups, and reduce the original problem to equation solvability over the underlying field. Further, we apply this new method to give a much more efficient algorithm for equation solvability over nilpotent rings than previously existed.

scholarly journals An Algorithm for Recognising the Exterior Square of a Multiset

2000 ◽  
Vol 3 ◽  
pp. 96-116 ◽  
Author(s):  
Catherine Greenhill
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Combinatorial Construction ◽  
Exterior Square

AbstractThe exterior square of a multiset is a natural combinatorial construction which is related to the exterior square of a vector space. We consider multisets of elements of an abelian group. Two properties are defined which a multiset may satisfy: recognisability and involution-recognisability. A polynomial-time algorithm is described which takes an input multiset and returns either (a) a multiset which is either recognisable or involution-recognisable and whose exterior square equals the input multiset, or (b) the message that no such multiset exists. The proportion of multisets which are neither recognisable nor involution-recognisable is shown to be small when the abelian group is finite but large. Some further comments are made about the motivating case of multisets of eigenvalues of matrices.

scholarly journals The extended equivalence and equation solvability problems for groups

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Gabor Horvath ◽  
Csaba Szabo
Keyword(s):  
Polynomial Time ◽  
Equivalence Problem ◽  
Nilpotent Groups ◽  
60Th Birthday ◽  
Special Issue ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Solvability Problem ◽  
Np Complete ◽  
Equation Solvability

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience We prove that the extended equivalence problem is solvable in polynomial time for finite nilpotent groups, and coNP-complete, otherwise. We prove that the extended equation solvability problem is solvable in polynomial time for finite nilpotent groups, and NP-complete, otherwise.

scholarly journals On Tensor-Factorisation Problems,I: The Combinatorial Problem

2004 ◽  
Vol 7 ◽  
pp. 73-100
Author(s):  
Peter M. Neumann ◽  
Cheryl E. Praeger
Keyword(s):  
Abelian Group ◽  
Tensor Product ◽  
Polynomial Time ◽  
Upper Bound ◽  
Combinatorial Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Probability Of Failure ◽  
Division Problem ◽  
Group A

AbstractA k-multiset is an unordered k-tuple, perhaps with repetitions. If x is an r-multiset {x1, …, xr} and y is an s-multiset {y1, …, ys} with elements from an abelian group A the tensor product x ⊗ y is defined as the rs-multiset {xi yj | 1 ≤ i ≤ r, 1 ≤ j ≤ s}. The main focus of this paper is a polynomial-time algorithm to discover whether a given rs-multiset from A can be factorised. The algorithm is not guaranteed to succeed, but there is an acceptably small upper bound for the probability of failure. The paper also contains a description of the context of this factorisation problem, and the beginnings of an attack on the following division-problem: is a given rs-multiset divisible by a given r-multiset, and if so, how can division be achieved in polynomially bounded time?

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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