Finding surfaces in simplicial complexes with bounded-treewidth 1-skeleton

07/21/2021

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by   Mitchell Black, et al.

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We consider the problem 2-Dim-Bounding-Surface. 2-Dim-Bounded-Surface asks whether or not there is a subcomplex S of a simplicial complex K homeomorphic to a given compact, connected surface bounded by a given subcomplex B⊂ K. 2-Dim-Bounding-Surface is NP-hard. We show it is fixed-parameter tractable with respect to the treewidth of the 1-skeleton of the simplicial complex K. Using some of the techniques we developed for the 2-Dim-Bounded-Surface problem, we obtain fixed parameter tractable algorithms for other topological problems such as computing an optimal chain with a given boundary and computing an optimal chain in a given homology class.