Boundary optimization for rough sets

TL;DR

This paper investigates how to optimize boundary regions in rough set theory by analyzing partitions of a set to minimize or maximize expected boundary size under different probability distributions.

Contribution

It introduces a novel boundary optimization framework for rough sets, reducing the problem to integer partition optimization and analyzing the weight function's shape.

Findings

Identifies partitions minimizing and maximizing expected boundary size.

Proves concave-convex properties of the weight function.

Establishes an analog to the AZ-identity in one case.

Abstract

Let $n > m\ge 2$ be integers and let $\mathcal{A}=\{A_1,\dots,A_m\}$ be a partition of $[n]=\{1,\dots,n\}$. For $X \subseteq [n]$, its $\mathcal{A}$-boundary region $\mathcal{A}(X)$ is defined to be the union of those blocks $A_i$ of $\mathcal{A}$ for which $A_i\cap X\neq \emptyset$ and $A_i\cap ([n] \setminus X)\neq \emptyset$. For three different probability distributions on the power set of $[n]$, partitions $\mathcal{A}$ of $[n]$ are determined such that the expected cardinality of the $\mathcal{A}$-boundary region of a randomly chosen subset of $[n]$ is minimal and maximal, respectively. The problem can be reduced to an optimization problem for integer partitions of $n$. In the most difficult case, the concave-convex shape of the corresponding weight function as well as several other inequalities are proved using an integral representation of the weight function. In one case, there is an interesting analogon to the AZ-identity. The study is motivated by the rough set theory.