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.

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 Applications of Generating Functions to Stochastic Processes and to the Complexity of the Knapsack Problem

2021 ◽  
Author(s):  
Jorma Jormakka ◽  
Sourangshu Ghosh
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Polynomial Time ◽  
Knapsack Problem ◽  
Generating Functions ◽  
General Theorem ◽  
Main Idea ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Linear Transform

The paper describes a method of solving some stochastic processes using generating functions. A general theorem of generating functions of a particular type is derived. A generating function of this type is applied to a stochastic process yielding polynomial time algorithms for certain partitions. The method is generalized to a stochastic process describing a rather general linear transform. Finally, the main idea of the method is used in deriving a theoretical polynomial time algorithm to the knapsack problem.

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.

Efficient Processing of XML Tree Pattern Queries

2006 ◽  
Vol 10 (5) ◽  
pp. 738-743 ◽  
Author(s):  
Yangjun Chen ◽  
◽  
Dunren Che ◽  
Keyword(s):  
Dynamic Programming ◽  
Polynomial Time ◽  
Main Idea ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Tree Pattern ◽  
Efficient Processing

In this paper, we present a polynomial-time algorithm for TPQ (tree pattern queries) minimization without XML constraints involved. The main idea of the algorithm is a dynamic programming strategy to find all the matching subtrees within a TPQ. A matching subtree implies a redundancy and should be removed in such a way that the semantics of the original TPQ is not damaged. Our algorithm consists of two parts: one for subtree recognization and the other for subtree deletion. Both of them needs only O(<I>n</I>2) time, where <I>n</I> is the number of nodes in a TPQ.

scholarly journals Non-commutative lattice problems

2016 ◽  
Vol 19 (3) ◽  
Author(s):  
Alexei Myasnikov ◽  
Andrey Nikolaev ◽  
Alexander Ushakov
Keyword(s):  
Computational Complexity ◽  
Large Class ◽  
Polynomial Time ◽  
Coxeter Groups ◽  
Nilpotent Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Group Element ◽  
Surface Groups ◽  
Nontrivial Element

AbstractWe 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 nontrivial 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.

The equation solvability problem over supernilpotent algebras with Mal’cev term

2018 ◽  
Vol 28 (06) ◽  
pp. 1005-1015 ◽  
Author(s):  
Michael Kompatscher
Keyword(s):  
Nilpotent Groups ◽  
Solvability Problem ◽  
Np Complete ◽  
Equation Solvability ◽  
Nilpotent Algebras

In 2011, Horváth gave a new proof that the equation solvability problem over finite nilpotent groups and rings is in P. In the same paper, he asked whether his proof can be lifted to nilpotent algebras in general. We show that this is in fact possible for supernilpotent algebras with a Mal’cev term. However, we also describe a class of nilpotent, but not supernilpotent algebras with Mal’cev term that have co-NP-complete identity checking problems and NP-complete equation solvability problems. This proves that the answer to Horváth’s question is negative in general (assuming P[Formula: see text]NP).

scholarly journals Distortion of embeddings of a torsion-free finitely generated nilpotent group into a unitriangular group

2017 ◽  
Vol 27 (06) ◽  
pp. 633-653
Author(s):  
Funda Gul ◽  
Alexei G. Myasnikov ◽  
Mahmood Sohrabi
Keyword(s):  
Heisenberg Group ◽  
Nilpotent Group ◽  
Polynomial Time ◽  
Nilpotent Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Finitely Generated ◽  
Unitriangular Group ◽  
Torsion Free

In this paper, we study distortion of various well-known embeddings of finitely generated torsion-free nilpotent groups [Formula: see text] into unitriangular groups [Formula: see text]. In particular, we show that there is no undistorted embeddings of [Formula: see text]-dimensional Heisenberg group into [Formula: see text]. We also provide a polynomial time algorithm for finding distortion of a given subgroup of [Formula: see text].

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.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

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