" In this chapter, we introduce some of the very basics that are used throughout the book.
First, we give the definition of a topological space and related notions of open and closed …

Cycle representatives of persistent homology classes can be used to provide descriptions of
topological features in data. However, the non-uniqueness of these representatives creates …

In this paper, we study the shape reconstruction problem, when the shape we wish to
reconstruct is an orientable smooth d-dimensional submanifold of the Euclidean space …

Metric spaces $(X, d) $ are ubiquitous objects in mathematics and computer science that
allow for capturing (pairwise) distance relationships $ d (x, y) $ between points $ x, y\in X …

Computing an optimal cycle in a given homology class, also referred to as the homology
localization problem, is known to be an NP-hard problem in general. Furthermore, there is …

Metric spaces (X, d) are ubiquitous objects in mathematics and computer science that allow
for capturing pairwise distance relationships d (x, y) between points x, y∈ X. Because of this …

Finding a cycle of lowest weight that represents a homology class in a simplicial complex is
known as homology localization (HL). Here we address this NP-complete problem using …

Cut problems form one of the most fundamental classes of problems in algorithmic graph
theory. In this paper, we initiate the algorithmic study of a high-dimensional cut problem. The …

The persistence barcode is a topological descriptor of data that plays a fundamental role in
topological data analysis. Given a filtration of data, the persistence barcode tracks the …

Homology features of spaces which appear in applications, for instance 3D meshes, are
among the most important topological properties of these objects. Given a non-trivial cycle in …