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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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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).

Cite AsGet BibTex

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

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.

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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