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Persistence Meets Resistance: Doubling down on Hardness

Authors Benedikt Kolbe , Tim Mayr

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LIPIcs.ICALP.2026.131.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Persistent homology
  • approximations
  • lower bounds
  • matrix rank
  • doubling dimension
  • Vietoris-Rips complex
  • Čech complex
  • measure bifiltration
  • subdivision-Rips bifiltration
  • Rhomboid filtration

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Abstract

We present results on the approximate computation of stable invariants for filtrations of finite metric spaces in the context of persistent homology. We establish novel approximation algorithms in the setting of n-point metric spaces where the growth of the doubling dimension is in o(log n) and the diameter is bounded. In the 1-parameter case, by revisiting known techniques (greedy permutations) in a new way, we derive the first linear-time algorithms for the problem of computing additive ε-approximations of any stable barcode. By deriving bounds on the convergence rate and the approximation quality of uniform samples, we extend the approach to selected multiparameter filtrations. We show that for normalized measure bifiltrations, including the multicover and subdivision-Rips bifiltration, any stable invariant can be probabilistically approximated in time constant in n. The constants in the running times of our algorithms depend on the doubling dimension, the diameter and the success probability. 
We further study the problem through the lens of fine-grained complexity and show that computing the rank of a matrix reduces to that of approximating the barcode of the Vietoris-Rips or Čech filtration. We present two variants of the reduction, one for sufficiently good additive approximations and the other for any constant factor multiplicative approximations.

Cite As Get BibTex

Benedikt Kolbe and Tim Mayr. Persistence Meets Resistance: Doubling down on Hardness. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 131:1-131:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.131

Author Details

Benedikt Kolbe
  • Hausdorff Center for Mathematics, University of Bonn, Germany
  • Lamarr Institute for Machine Learning and Artificial Intelligence, University of Bonn, Germany
Tim Mayr
  • University of Bonn, Germany

Funding

  • Kolbe, Benedikt: This work was partially supported by the Lamarr Institute for Machine Learning and Artificial Intelligence.

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