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ETH-Tight Algorithms for Finding Surfaces in Simplicial Complexes of Bounded Treewidth

Authors Mitchell Black, Nello Blaser , Amir Nayyeri, Erlend Raa Vågset

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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
  • Computational Geometry
  • Surface Recognition
  • Treewidth
  • Hasse Diagram
  • Simplicial Complexes
  • Low-Dimensional Topology
  • Parameterized Complexity
  • Computational Complexity

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Abstract

Given a simplicial complex with n simplices, we consider the Connected Subsurface Recognition (c-SR) problem of finding a subcomplex that is homeomorphic to a given connected surface with a fixed boundary. We also study the related Sum-of-Genus Subsurface Recognition (SoG) problem, where we instead search for a surface whose boundary, number of connected components, and total genus are given. For both of these problems, we give parameterized algorithms with respect to the treewidth k of the Hasse diagram that run in 2^{O(k log k)}n^{O(1)} time. For the SoG problem, we also prove that our algorithm is optimal assuming the exponential-time hypothesis. In fact, we prove the stronger result that our algorithm is ETH-tight even without restriction on the total genus.

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Mitchell Black, Nello Blaser, Amir Nayyeri, and Erlend Raa Vågset. ETH-Tight Algorithms for Finding Surfaces in Simplicial Complexes of Bounded Treewidth. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.17

Author Details

Mitchell Black
  • School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA
Nello Blaser
  • Department of Informatics, University of Bergen, Norway
Amir Nayyeri
  • School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA
Erlend Raa Vågset
  • Department of Informatics, University of Bergen, Norway

Funding

  • Black, Mitchell: This author was supported in part by NSF grants CCF-1941086 and CCF-1816442.
  • Nayyeri, Amir: This author was supported in part by NSF grants CCF-1941086 and CCF-1816442.
  • Vågset, Erlend Raa: This author was supported in part by the Research Council of Norway grant “Parameterized Complexity for Practical Computing (PCPC)” (NFR, no. 274526).

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