Computer Science > Computational Geometry
[Submitted on 21 Jul 2021 (v1), last revised 16 Mar 2022 (this version, v3)]
Title:Finding minimum bounded and homologous chains in simplicial complexes with bounded-treewidth 1-skeleton
View PDFAbstract: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 $K$.
Submission history
From: Mitchell Black [view email][v1] Wed, 21 Jul 2021 20:09:35 UTC (948 KB)
[v2] Sun, 19 Dec 2021 21:36:24 UTC (952 KB)
[v3] Wed, 16 Mar 2022 16:34:03 UTC (962 KB)
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