# Computing Minimal Persistent Cycles: Polynomial and Hard Cases

## TL;DR

This paper investigates the computational complexity of finding minimal persistent cycles across different dimensions, proving NP-hardness in general but identifying specific tractable cases related to weak pseudomanifolds.

## Contribution

It extends the understanding of minimal persistent cycle computation to higher dimensions and introduces polynomial algorithms for certain cases involving weak pseudomanifolds.

## Key findings

- NP-hardness for d>1 persistent cycles in general complexes
- Polynomial algorithms for finite intervals in weak pseudomanifolds
- Experiments show minimal cycles capture significant data features

## Abstract

Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in a purely topological persistence diagram (also termed as barcode). In our earlier work, we showed that computing minimal 1-dimensional persistent cycles (persistent 1-cycles) for finite intervals is NP-hard while the same for infinite intervals is polynomially tractable. In this paper, we address this problem for general dimensions with $\mathbb{Z}_2$ coefficients. In addition to proving that it is NP-hard to compute minimal persistent d-cycles (d>1) for both types of intervals given arbitrary simplicial complexes, we identify two interesting cases which are polynomially tractable. These two cases assume the complex to be a certain generalization of manifolds which we term as weak pseudomanifolds. For finite intervals from the d-th persistence diagram of a weak (d+1)-pseudomanifold, we utilize the fact that persistent cycles of such intervals are null-homologous and reduce the problem to a minimal cut problem. Since the same problem for infinite intervals is NP-hard, we further assume the weak (d+1)-pseudomanifold to be embedded in $\mathbb{R}^{d+1}$ so that the complex has a natural dual graph structure and the problem reduces to a minimal cut problem. Experiments with both algorithms on scientific data indicate that the minimal persistent cycles capture various significant features of the data.