Finding minimum bounded and homologous chains in simplicial complexes with bounded-treewidth 1-skeleton

Mitchell Black; Amir Nayyeri

Finding minimum bounded and homologous chains in simplicial complexes with bounded-treewidth 1-skeleton

Computational Geometry 2022-03-17 v3

Authors: Mitchell Black , Amir Nayyeri

Abstract

We consider two problems on simplicial complexes: the Optimal Bounded Chain Problem and the Optimal Homologous Chain Problem. The Optimal Bounded Chain Problem asks to find the minimum weight $d$ -chain in a simplicial complex $K$ bounded by a given $(d{-}1)$ -chain, if such a $d$ -chain exists. The Optimal Homologous Chain problem asks to find the minimum weight $(d{-}1)$ -chain in $K$ homologous to a given $(d{-}1)$ -chain. Both of these problems are NP-hard and hard to approximate within any constant factor assuming the Unique Games Conjecture. We prove that these problems are fixed-parameter tractable with respect to the treewidth of the 1-skeleton of $K$ .

Keywords

approximation algorithm

Cite

@article{arxiv.2107.10339,
  title  = {Finding minimum bounded and homologous chains in simplicial complexes with bounded-treewidth 1-skeleton},
  author = {Mitchell Black and Amir Nayyeri},
  journal= {arXiv preprint arXiv:2107.10339},
  year   = {2022}
}

Comments

In v2, we replaced an incorrect theorem in Section 7. Versions v1 and v2 contained an algorithm for finding subcomplexes that are homeomorphic to surfaces in simplicial complexes. In version v3, this material has been moved to arXiv:2203.07566. The split into two papers reflects new lower bounds on finding surfaces and a change in authorship

Finding minimum bounded and homologous chains in simplicial complexes with bounded-treewidth 1-skeleton

Abstract

Keywords

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