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ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes

Authors Geevarghese Philip , Erlend Raa Vågset

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LIPIcs.SoCG.2026.85.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Algorithm design techniques
  • Mathematics of computing → Combinatorial algorithms
  • Mathematics of computing → Algebraic topology
Keywords
  • Discrete Morse Theory
  • Simplicial Complexes
  • Optimal Morse Matching
  • Treewidth
  • Parameterized Algorithms
  • Computational Topology
  • Dynamic Programming
  • Exponential Time Hypothesis
  • Topological Data Analysis

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Abstract

The Optimal Morse Matching (OMM) problem asks for a discrete gradient vector field on a simplicial complex that minimizes the number of critical simplices. It is NP-hard and has been studied extensively in heuristic, approximation, and parameterized complexity settings. Parameterized by treewidth k, OMM has long been known to be solvable on triangulations of 3-manifolds in 2^O(k²) n^O(1) time and in FPT time for triangulations of arbitrary manifolds, but the exact dependence on k has remained an open question. We resolve this by giving a new 2^O(k log k) n-time algorithm for any finite regular CW complex, and show that no 2^o(k log k) n^O(1)-time algorithm exists unless the Exponential Time Hypothesis (ETH) fails.

Cite As Get BibTex

Geevarghese Philip and Erlend Raa Vågset. ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 85:1-85:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.85

Author Details

Geevarghese Philip
  • Chennai Mathematical Institute, India
  • CNRS IRL ReLaX, Chennai, India
Erlend Raa Vågset
  • Western Norway University of Applied Sciences (HVL), Førde, Norway

Funding

  • Vågset, Erlend Raa: Supported in part by the Research Council of Norway, grant “Parameterized Complexity for Practical Computing (PCPC)” (No. 274526).

Acknowledgements

Geevarghese Philip thanks Priyavrat Deshpande for introducing him to Morse matchings and to questions concerning their efficient computation.

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