ETH-tight algorithms for finding surfaces in simplicial complexes of bounded treewidth
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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.07566
- arXiv:
- arXiv:2203.07566
- Bibcode:
- 2022arXiv220307566B
- Keywords:
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- Computer Science - Computational Geometry
- E-Print:
- This paper contains some material that previously appeared at arXiv:2107.10339. The split into two papers reflects new material and a change in authorship in this version. Accepted to SoCG 2022