This paper is a survey on discrete isoperimetric type problems. We present here as some known facts about their solutions as well some new results and demonstrate a general techniques used in this area. The main attention is paid to the unit cube and cube like structures. Besides some applications of the isoperimetric approach are listed too. 1 Introduction This paper is devoted to the discrete isoperimetric problem. This problem may be considered as an analog of the well known continuous problem and has some similar features. The discrete isoperimetric problem began to be studied a very long ago and a lot about it's solutions is known now. If there is the only solution of the continuous version, the discrete one, considered for the unit cube, has generally more solutions with much more rich structure, which have no direct continuous analogs. It is mainly due to the facts that, at first not for all values of cardinality of a subset (which is defined as the number of cube points in the...

Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.

In this paper, we prove two general theorems on monotone Boolean functions which are useful for constructing an learning algorithm for monotone Boolean functions under the uniform distribution. A monotone

In the context of mining for frequent patterns using the standard levelwise algorithm, the following question arises: given the current level and the current set of frequent patterns, what is the maximal number of candidate patterns that can be generated on the next level? We answer this question by providing a tight upper bound, derived from a combinatorial result from the sixties by Kruskal and Katona. Our result is useful to reduce the number of database scans.

this paper we solve the Shadow Minimization Problem for the poset of submatrices of a matrix, i.e. we prove a theorem for this poset which is analogous to the classical Kruskal--Katona theorem [7, 8] for Boolean lattices. For all definitions not included in this article we refer to Engel's book [2]. 1.1. Submatrix Orders. Let M be a matrix and R and C its sets of rows and columns, respectively. Clearly, every nonempty submatrix L of M can be considered as a pair (S; T ), where ; 6= S ` R; ; 6= T ` C and L consists of all cells lying in a row from S

We consider a variant of what is known as the discrete isoperimetric problem, namely the problem of minimising the size of the boundary of a family of subsets of a nite set. We use the technique of `shifting' to provide an alternative proof of a result of Hart. This technique was introduced in the early 1980s by Frankl and Furedi and gave alternative proofs of previously known classical results like the discrete isoperimetric problem itself and the Kruskal-Katona theorem. Hence our purpose is to bring Hart's result into this general framework.

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