Variational Principle in Discrete Extremal Problems

The paper is devoted to a new approach for solution of some discrete extremal problems, such as the edge isoperimetric problem for graphs, the bisection width problem, the shadow and boundary minimization problems and others, on the carte-sian product of graphs and posets. The main aspect of the approach is in nding of some equivalence relations between the functions in the named problems via reducing these problems to the construction of extremal ideals in the cartesian product of chains or some other related posets. If some two problems are reducible to the same extremal ideals problem, they are in a sense equivalent. This techniques allows in particular to replace an original problem for a rather complicated structure to some known (resp. solved) problem from its equivalence class on a simpler structure, and to get solutions for the original problem based on the solution of the simpliied problem. As direct consequences of our approach we solve the edge isoperimetric problem for the cartesian product of arbitrary trees and show that this problem itself is a consequence of the Kruskal-Katona type problem for the partial mappings semilat-tice 6]. We give a short survey in the listed areas and present new results in the frameworks of the approach.