Volume 14, Issue 6

We use randomised rounding to obtain an upper bound for the optimum value of the program {min cx | A x ≥ b , x ≥ 0 , x is an integer vector}, where b > 0, c ≥ 0 are rational vectors and A is an arbitrary rational matrix. Our bound generalises some known bounds for covering integer programs (that is, the same programs with the restriction that all elements of A are non-negative).

We introduce the notion of a homomorphic relation of algebras with a scheme of operators. Homomorphisms and π -homomorphisms are special cases of this notion. Some new relations of algebras with a scheme of operators are studied.

We consider the problem of finding connections between eigen-vectors and subgraphs of a weighted undirected graph G . Let G have n vertices labelled 1, . . . , n , λ be an eigen-value of the graph G of multiplicity t ≥ 1, and let X (i) = (, . . . , ), i = 1, . . . , t , be linearly independent eigen-vectors corresponding to this eigen-value. We obtain formulas representing the components of the eigen-vectors X (i) in terms of some characteristics of special subgraphs of the graph G , i = 1, . . . , t , j = 1, . . . , n . An illustrative example is given.

For a problem of interval search on the Boolean cube, we study the asymptotic behaviour of the average search time in the class of balanced tree circuits on sequences of positive integers { k i } under the assumption that k i are cardinalities of databases. We show that the asymptotic behaviour may differ on different sequences. We give a complete description of the class of possible behaviours.

We suggest a canonical system of defining relations for finite everywhere defined output-less automata. We construct a procedure to pass from an arbitrary finite system of defining relations to a canonical one and, as a corollary, a procedure to check whether a finite system of pairs of words is a defining system for a given automaton or not. We also suggest a procedure to pass from a traversal of all arcs of the automaton graph to a system of defining relations and vice versa.

We consider free (not rooted) trees with n labelled vertices whose multiplicities take values in some fixed subset A of non-negative integers such that A contains zero, A ≠ {0}, A ≠ {0, 1}, and the greatest common divisor of the numbers { k | k ∈ A } is equal to one. We find the asymptotic behaviour of the number of all these trees as n → ∞. Under the assumption that the uniform distribution is defined on the set of these trees, for the random variable , r ∈ A , which is equal to the number of vertices of multiplicity r in a randomly chosen tree, we find the asymptotic behaviour of the mathematical expectation and variance as n → ∞ and prove local normal and Poisson theorems for these random variables. For the case A = {0,1}, we obtain estimates of the number of all forests with n labelled vertices consisting of N free trees as n → ∞ under various constraints imposed on the function N = N(n) . We find the asymptotic behaviour of the number of all forests of free trees with n vertices of multiplicities at most one. We prove local normal and Poisson theorems for the number of trees of given size and for the total number of trees in a random forest of this kind. We obtain limit distribution of the random variable equal to the size of the tree containing the vertex with given label.

We consider the class of Boolean functions F n,k consisting of all functions in n variables such that each of them takes value one exactly for k tuples of variables. We obtain linear in n estimates of the complexity of realisation of functions in F n,k by circuits of functional elements over the basis containing all Boolean functions in two variables except the linear functions x ⊕ y and x ⊕ y ⊕ 1. It follows from these estimates that for small k , for example, for k < ln n , the well-known Finikov method provides asymptotically minimal circuits for all functions of F n,k . In some cases, the known lower bounds for complexity of circuits give a possibility to prove the minimality of the corresponding circuits.

We suggest an algorithm to solve a two-dimensional interval search problem which is characterised by the memory consumed of order k , the average search time (without the time needed to output the answer) of order √ k , where k is the database size.