On covering problems in hierarchical poset spaces

Abstract

Given a ground set $ \mathcal{U} $ and a collection $ \mathcal{C} $ of subsets of $ \mathcal{U} $, we investigate the covering number, which is the cardinality of the smallest sub-collection $ \mathcal{S} $ whose union covers the universe $ \mathcal{U} $. This paper embraces three covering-related problems in hierarchical poset spaces: the classic, the short covering and normal codes. In the first two problems, the classical and the short-covering ones, the covering numbers are studied for collections of $ R $-balls and $ R $-strips, respectively. We prove that determining the covering number in hierarchical spaces depends only on their correspondents in the Hamming space. It allows us to express the covering number in hierarchical spaces. Later on, we characterize all $ \mathcal{H} $-normal codes in binary hierarchical spaces, that are a distinctive family of $ R $-coverings.