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 the minimal distance of a polynomial code

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Peter Pal Pach ◽  
Csaba Szabo
Keyword(s):  
Finite Subset ◽  
60Th Birthday ◽  
Theoretical Problem ◽  
Minimal Distance ◽  
Distribution Of Primes ◽  
Special Issue ◽  
Theoretical Methods ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Special Case

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For a polynomial f(x) is an element of Z(2)[x] it is natural to consider the near-ring code generated by the polynomials f circle x, f circle x(2) ,..., f circle x(k) as a vectorspace. It is a 19 year old conjecture of Gunter Pilz that for the polynomial f (x) - x(n) broken vertical bar x(n-1) broken vertical bar ... broken vertical bar x the minimal distance of this code is n. The conjecture is equivalent to the following purely number theoretical problem. Let (m) under bar = \1, 2 ,..., m\ and A subset of N be an arbitrary finite subset of N. Show that the number of products that occur odd many times in (n) under bar. A is at least n. Pilz also formulated the conjecture for the special case when A = (k) under bar. We show that for A = (k) under bar the conjecture holds and that the minimal distance of the code is at least n/(log n)(0.223). While proving the case A = (k) under bar we use different number theoretical methods depending on the size of k (respect to n). Furthermore, we apply several estimates on the distribution of primes.

scholarly journals Separating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Thomas P. Hayes
Keyword(s):  
Positive Integer ◽  
Open Problem ◽  
Communication Complexity ◽  
60Th Birthday ◽  
Equal Size ◽  
Special Issue ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Zarankiewicz Problem ◽  
Complexity Hierarchy

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For every positive integer k, we construct an explicit family of functions f : \0, 1\(n) -\textgreater \0, 1\ which has (k + 1) - party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Omega (n/k(4)2(k)) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain K-s,K-t as a subgraph.

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 On the metric dimension of Grassmann graphs

2012 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Robert F. Bailey ◽  
Karen Meagher
Keyword(s):  
Upper Bound ◽  
60Th Birthday ◽  
Metric Dimension ◽  
Special Issue ◽  
Resolving Set ◽  
Grassmann Graph ◽  
Algorithms And Complexity ◽  
International Audience

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience The metric dimension of a graph Gamma is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph G(q)(n, k) (whose vertices are the k-subspaces of F-q(n), and are adjacent if they intersect in a (k 1)-subspace) for k \textgreater= 2. We find an upper bound on its metric dimension, which is equal to the number of 1-dimensional subspaces of F-q(n). We also give a construction of a resolving set of this size in the case where k + 1 divides n, and a related construction in other cases.

scholarly journals Computing tensor decompositions of finite matrix groups

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Nurullah Ankaralioglu ◽  
Akos Seress
Keyword(s):  
Tensor Decomposition ◽  
60Th Birthday ◽  
Special Issue ◽  
Matrix Groups ◽  
Tensor Decompositions ◽  
Products Of Groups ◽  
Finite Matrix ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Central Products

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience We describe an algorithm to compute tensor decompositions of central products of groups. The novelty over previous algorithms is that in the case of matrix groups that are both tensor decomposable and imprimitive, the new algorithm more often outputs the more desirable tensor decomposition.

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 On the sensitivity of cyclically-invariant Boolean functions

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Sourav Chakraborty
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Transitive Group ◽  
60Th Birthday ◽  
Special Issue ◽  
Natural Class ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Class Of Functions ◽  
Block Sensitivity

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience In this paper we construct a cyclically invariant Boolean function whose sensitivity is Theta(n(1/3)). This result answers two previously published questions. Turan (1984) asked if any Boolean function, invariant under some transitive group of permutations, has sensitivity Omega(root n). Kenyon and Kutin (2004) asked whether for a "nice" function the product of 0-sensitivity and 1-sensitivity is Omega(n). Our function answers both questions in the negative. We also prove that for minterm-transitive functions (a natural class of Boolean functions including our example) the sensitivity is Omega(n(1/3)). Hence for this class of functions sensitivity and block sensitivity are polynomially related.

scholarly journals On the residual solvability of generalized free products of solvable groups

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Delaram Kahrobaei ◽  
Stephen Majewicz
Keyword(s):  
Free Product ◽  
Solvable Groups ◽  
60Th Birthday ◽  
Special Issue ◽  
Free Products ◽  
Generalized Free Product ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Generalized Free Products

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience In this paper, we study the residual solvability of the generalized free product of solvable groups.

scholarly journals Combinatorics, Groups, Algorithms, and Complexity: Conference in honor of Laci Babai's 60th birthday

2010 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Akos Seress ◽  
Mario Szegedy
Keyword(s):  
Theoretical Computer Science ◽  
Ohio State University ◽  
60Th Birthday ◽  
State University ◽  
Special Issue ◽  
Theoretical Computer ◽  
Introductory Article ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Generous Support

International audience Honoring László (Laci) Babai's 60th birthday, the conference "Combinatorics, Groups, Algorithms, and Complexity" (Ohio State University, March 15-25, 2010) explored the links between the areas mentioned in the title. These areas represent Laci's wide interests in mathematics and theoretical computer science; his work has revealed and enriched many of the interconnections between them. The conference had 109 participants from North America, Europe, Asia, and Australia (31 of them from overseas), including 3 Nevanlinna prize winners, 32 students, 13 postdocs, 20 females, and 18 former and current students of Laci Babai. The program consisted of 73 talks and a problem session. The full list of talks can be found in the introductory article by the guest editors of this special issue who also served as the organizers of the conference. We thank all participants and speakers for the success of the conference. We wish to express our gratitude to the National Science Foundation, National Security Agency, and The Ohio State Mathematical Research Institute for their generous support. This special issue contains papers in the conference topics, but not necessarily coinciding with the authors' talks at the conference. Each paper has been peer-reviewed. Toniann Pitassi, László Pyber, Uwe Schöning, Jiří Sgall, and Aner Shalev served with us as editors of this special issue. We thank for their work as well as for the assistance of the anonymous referees.

scholarly journals On paths, trails and closed trails in edge-colored graphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Laurent Gourvès ◽  
Adria Lyra ◽  
Carlos A. Martinhon ◽  
Jérôme Monnot
Keyword(s):  
Polynomial Time ◽  
Optimal Solution ◽  
Colored Graph ◽  
Edge Disjoint ◽  
Colored Graphs ◽  
International Audience ◽  
Np Complete ◽  
T Path ◽  
Vertex Disjoint

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

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