Volume 3, Issue 4

For the past decade the Steiner minimal tree problem has attracted the attention of researchers in discrete optimization. A brief survey of the main results concerning the properties and algorithms of the Steiner problem in the Euclidean plane, the Steiner problem in the plane with rectilinear metric and the Steiner problem in networks is done in this paper. The main attention is paid to the recent results concerning the last problem.

A square non-negative matrix is called primitive if all the elements of some power of this matrix are positive. The exponent of a primitive matrix is the minimum power satisfying this condition. The exponent of a class of primitive matrices is the minimum power such that all the matrices of the class to this power have positive elements only. In this paper, bounds for the exponents of primitive matrices and classes of matrices are obtained.

Some subgroups of automorphisms of Grassmann's algebra and their explicit descriptions are considered. The notion of a graduated determinant is introduced and its properties are investigated.

We prove Brenner-Evans' conjecture that, for any natural numbers k ≥ 4 and m ≥ 1, any even permutation of the group A km is a product of two permutations, each of them is expanded into a product of m independent cycles of length k. It is known that this assertion is incorrect for k = 2, 3.

The notion of the line hypergraph is introduced. It is an immediate generalization of two wellknown objects: a line graph and a dual hypergraph. We obtain various characterizations of line hypergraphs; we also obtain a generalization of Whitney's theorem. The NP-completeness of the problem of determining whether a given graph is the line graph of a hypergraph of rank r > 2 is proved.

A Boolean function is called a threshold function if its truth domain is a part of the n-cube cut off by some hyperplane. The number of threshold functions of n variables P(2, n) was estimated in [1, 2, 3]. Obtaining the lower bounds is a problem of special difficulty. Using a result of the paper [4], Zuev in [3] showed that for sufficiently large n P(2, n) > 2 n²(1-10/ln n). In the present paper a new proof which gives a more precise lower bound of P(2, n) is proposed, namely, it is proved that for sufficiently large n P(2, n) > 2 n²(1-7/ln n) P(2, [(7(n-1)ln2)/ln(n- 1)]).

The paper is devoted to the relation between random trees and branching processes for which the number of descendants of one particle has the Poisson distribution. The main result of the paper consists in finding the leading term of the asymptotics of the probabilities of large deviations for the height of a random tree. Similar problems were considered in [1-6].

Theorems on large deviations in the polynomial scheme of trials with a fixed number of outcomes and a growing number of trials, which generalize the analogous known results for the case of binomial scheme, are obtained.