On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class

TL;DR

This paper investigates the complexity of finding optimal cycles in homology classes, providing efficient algorithms for some cases and proving hardness results for others, especially in 3-manifolds.

Contribution

It introduces a simple algorithm for lexicographic optimal cycles in 1-homology classes and establishes computational hardness for bottleneck optimal cycles in 3-manifolds.

Findings

Efficient algorithm for lex-optimal cycles in 1-homology classes.

Hardness results for bottleneck optimal cycles in 3-manifolds.

Distinction between persistent homology and ordinary homology computation complexities.

Abstract

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 a homology class, we consider the problem of computing a representative in that homology class which is optimal. We study two measures of optimality, namely, the lexicographic order of cycles (the lex-optimal cycle) and the bottleneck norm (a bottleneck-optimal cycle). We give a simple algorithm for computing the lex-optimal cycle for a 1-homology lass in a closed orientable surface. In contrast to this, our main result is that, in the case of 3-Manifolds of size $n^2$ in the Euclidean 3-space, the problem of finding a bottleneck optimal cycle cannot be solved more efficiently than solving a system of linear equations with an $n \times n$ sparse matrix. From this reduction, we deduce several hardness results. Most notably, we show that for 3-manifolds given as a subset of the 3-space of size $n^2$, persistent homology computations are at least as hard as rank computation (for sparse matrices) while ordinary homology computations can be done in $O(n^2 \log n)$ time. This is the first such distinction between these two computations. Moreover, it follows that the same disparity exists between the height persistent homology computation and general sub-level set persistent homology computation for simplicial complexes in the 3-space.