Discrete analysis and operations research

Reduction of a minimization problem of a separable convex function under linear constraints to a fixed point problem

2019-10-01T08:32:05+00:00

The paper is devoted to studying a constrained nonlinear optimization problem of a special kind. The objective functional of the problem is a separable convex function whose minimum is sought for on a set of linear constraints in the form of equalities. It is proved that, for this type of optimization problems, the explicit form can be obtained of a projection operator based on a generalized projection matrix. The projection operator allows us to represent the initial problem as a fixed point problem. The explicit form of the fixed point problem makes it possible to run a process of simple iteration. We prove the linear convergence of the obtained iterative method and, under rather natural additional conditions, its quadratic convergence. It is shown that an important application of the developed method is the flow assignment in a network of an arbitrary topology with one pair of source and sink.
Bibliogr. 10.

The Hamming distance spectrum between self-dual Maiorana–McFarland bent functions

2019-10-01T08:22:18+00:00

A bent function is self-dual if it is equal to its dual function. We study the metric properties of the self-dual bent functions constructed on using available constructions. We find the full Hamming distance spectrum between self-dual Maiorana–McFarland bent functions. Basing on this, we find the minimal Hamming distance between the functions under study.
Bibliogr. 22.

On the complexity of multivalued logic functions over some infinite basis

2019-10-01T08:28:06+00:00

Under study is the complexity of the realization of k-valued logic functions (k≥3) by logic circuits in the infinite basis consisting of the Post negation (i.e., the function (x+1) mod k) and all monotone functions. The complexity of the circuit is the total number of elements of this circuit. For an arbitrary function f, we find the lower and upper bounds of complexity which differ from one another at most by 1 and have the form 3log3 (d(f)+1)+O(1), where d(f) is the maximal number of the decrease of the value of f taken over all increasing chains of tuples of values of the variables. We find the exact value of the corresponding Shannon function which characterizes the complexity of the most complex function of a given number of variables.
Illustr. 4, bibliogr. 24.

Tree-like structure graphs with full diversity of balls

2019-10-01T08:45:49+00:00

Under study is the diversity of metric balls in connected finite ordinary graphs considered as a metric space with the usual shortest-path metric. We investigate the structure of graphs in which all balls of fixed radius i are distinct for each i less than the diameter of the graph. Let us refer to such graphs as graphs with full diversity of balls. For these graphs, we establish some properties connected with the existence of bottlenecks and find out the configuration of blocks in the graph. Using the obtained properties, we describe the tree-like structure graphs with full diversity of balls.
Illustr. 8, bibliogr. 22

Rectifier circuits of bounded depth

2019-10-01T08:45:50+00:00

Asymptotically tight bounds are obtained for the complexity of computation of the classes of $(m, n)$ -matrices with entries from the set $0, 1, \dots, q - 1$ by rectifier circuits of bounded depth $d,$ under some relations between $m$ , $n$ , and $q$ . In the most important case of $q = 2$ , it is shown that the asymptotics of the complexity of Boolean $(m, n)$ -matrices, log $n = o (m)$ , log $m = o (n)$ , is achieved for the circuits of depth 3.
Illustr. 1, bibliogr. 11.