Abstract

We consider the problem of finding a subcomplex $\mathcal{K}'$ of a simplicial complex $\mathcal{K}$ such that $\mathcal{K}'$ is homeomorphic to the 2-dimensional sphere, $\mathbb{S}^2$. We study two variants of this problem. The first asks if there exists such a $\mathcal{K}'$ with at most $\mathcal{K}$ triangles, and we show that this variant is ${\mathsf{W[1]}}$-hard and, assuming the exponential time hypothesis, admits no $n^{o(\sqrt{k})}$-time algorithm. We also give an algorithm that is tight with regard to this lower bound. The second problem is the dual of the first and asks if $\mathcal{K}'$ can be found by removing at most $k$ triangles from $\mathcal{K}$. This variant has an immediate $\mathcal{O}(3^{k}poly(|\mathcal{K}|))$-time algorithm, and we show that it admits a polynomial kernelization to $\mathcal{O}(k^2)$ triangles, as well as a polynomial compression to a weighted version with bit-size $\mathcal{O}(k \log k)$.

Keywords

  1. computational topology
  2. parameterized complexity
  3. simplicial complex

MSC codes

  1. 57M20
  2. 68Q17
  3. 68Q25

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Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 2092 - 2110
ISSN (online): 1095-7146

History

Submitted: 2 February 2018
Accepted: 31 July 2019
Published online: 29 October 2019

Keywords

  1. computational topology
  2. parameterized complexity
  3. simplicial complex

MSC codes

  1. 57M20
  2. 68Q17
  3. 68Q25

Authors

Affiliations

Funding Information

Javna Agencija za Raziskovalno Dejavnost RS https://doi.org/10.13039/501100004329 : P1–0297, L7–5459, J1–8130

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