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

Authors Erin Wolf Chambers, Salman Parsa, Hannah Schreiber

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Author Details

Erin Wolf Chambers
  • Saint Louis University, MO, USA
Salman Parsa
  • University of Utah, Salt Lake City, UT, USA
Hannah Schreiber
  • Saint Louis University, MO, USA

Funding

  • Chambers, Erin Wolf: This author was funded in part by the National Science Foundation through grants CCF-1614562, CCF-1907612, CCF-2106672, and DBI-1759807.
  • Parsa, Salman: This author was funded in part by the Saint Louis University Research Institute and by NSF grant CCF-1614562.
  • Schreiber, Hannah: This author was funded in part by the National Science Foundation through grant DBI-1759807.

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Erin Wolf Chambers, Salman Parsa, and Hannah Schreiber. On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.25

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 class in a closed orientable surface. In contrast to this, our main result is that, in the case of 3-manifolds of size n² 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 × 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 n², persistent homology computations are at least as hard as rank computation (for sparse matrices) while ordinary homology computations can be done in O(n² 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Mathematics of computing → Geometric topology
  • Theory of computation → Computational geometry
Keywords
  • computational topology
  • bottleneck optimal cycles
  • homology

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