Hasmik Sahakyan

Numerical characterization of n-cube subset partitioning

A general quantitative description of vertex subsets of the n-dimensional unit cube E^n through their partitions (direct problem) is given and the existence and composition problems for vertex subsets with given quantitative characteristics of partitions (inverse problem) are considered. Each of these subproblems is of significant theoretical and practical importance. Finding an efficient algorithmic solution to the inverse problem remains open. A complete and simple structural description of the numerical parameters of the unit cube partitions is presented.

The splitting technique in monotone recognition

We consider the problem of query based algorithmic identification/recognition of monotone Boolean functions, as well as of binary functions defined on multi-valued discrete grids. Hansels chain-split technique of n -cubes is a well known effective tool of monotone Boolean recognition. An extension by Alekseev is already applied to the grid case. The practical monotone recognition on n -cubes is provided by the so called chain-computation algorithms that is not extended to the case of multi-valued grids. We propose a novel split construction based on partitioning the grid into sub-grids and into discrete structures that are isomorphic to binary cubes. Monotonicity in a multi-valued grid implies monotonicity in all induced binary cubes and in multi-valued sub-grids. Applying Hansels technique for identification of monotone Boolean functions on all appearing binary cubes, and Alekseevs algorithm on all sub-grids leads to different scenarios of reconstruction of monotone functions. On one hand such partitioning technique makes parallel recognition possible, on the other hand - the method can be used in practical identification algorithms due to simple structures and easily calculable quantities appearing after the partition to the n -cubes. Complexity issues of considered algorithms were also elaborated.