Vol. 13 no. 4

Special Issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity

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

Seress, Akos ; Szegedy, Mario.
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 […]

2. Computing tensor decompositions of finite matrix groups

Ankaralioglu, Nurullah ; Seress, Akos.
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.

3. Separating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem

Hayes, Thomas P..
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. […]

4. The extended equivalence and equation solvability problems for groups

Horvath, Gabor ; Szabo, Csaba.
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.

5. On the minimal distance of a polynomial code

Pach, Peter Pal ; Szabo, Csaba.
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 […]

6. On the residual solvability of generalized free products of solvable groups

Kahrobaei, Delaram ; Majewicz, Stephen.
In this paper, we study the residual solvability of the generalized free product of solvable groups.

7. On the sensitivity of cyclically-invariant Boolean functions

Chakraborty, Sourav.
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 […]

8. Polynomial-time normalizers

Luks, Eugene M. ; Miyazaki, Takunari.
For an integer constant d \textgreater 0, let Gamma(d) denote the class of finite groups all of whose nonabelian composition factors lie in S-d; in particular, Gamma(d) includes all solvable groups. Motivated by applications to graph-isomorphism testing, there has been extensive study of the complexity of computation for permutation groups in this class. In particular, the problems of finding set stabilizers, intersections and centralizers have all been shown to be polynomial-time computable. A […]

9. On the metric dimension of Grassmann graphs

Bailey, Robert F. ; Meagher, Karen.
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 […]