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Minimum Bounded Chains and Minimum Homologous Chains in Embedded Simplicial Complexes

Authors Glencora Borradaile, William Maxwell, Amir Nayyeri

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Author Details

Glencora Borradaile
  • Oregon State University, Corvallis, OR, USA
William Maxwell
  • Oregon State University, Corvallis, OR, USA
Amir Nayyeri
  • Oregon State University, Corvallis, OR, USA

Funding

This material is based upon work supported by the National Science Foundation under Grant Nos. CCF-1617951 and CCF-1816442.

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Glencora Borradaile, William Maxwell, and Amir Nayyeri. Minimum Bounded Chains and Minimum Homologous Chains in Embedded Simplicial Complexes. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.21

Abstract

We study two optimization problems on simplicial complexes with homology over β„€β‚‚, the minimum bounded chain problem: given a d-dimensional complex 𝒦 embedded in ℝ^(d+1) and a null-homologous (d-1)-cycle C in 𝒦, find the minimum d-chain with boundary C, and the minimum homologous chain problem: given a (d+1)-manifold β„³ and a d-chain D in β„³, find the minimum d-chain homologous to D. We show strong hardness results for both problems even for small values of d; d = 2 for the former problem, and d=1 for the latter problem. We show that both problems are APX-hard, and hard to approximate within any constant factor assuming the unique games conjecture. On the positive side, we show that both problems are fixed-parameter tractable with respect to the size of the optimal solution. Moreover, we provide an O(√{log Ξ²_d})-approximation algorithm for the minimum bounded chain problem where Ξ²_d is the dth Betti number of 𝒦. Finally, we provide an O(√{log n_{d+1}})-approximation algorithm for the minimum homologous chain problem where n_{d+1} is the number of (d+1)-simplices in β„³.

Subject Classification

ACM Subject Classification
  • Theory of computation β†’ Computational geometry
Keywords
  • computational topology
  • algorithmic complexity
  • simplicial complexes

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