Allowing cycles in discrete Morse theory

Abstract : Discrete gradient vector fields are combinatorial structures that can be used for accelerating the homology computation of CW complexes, such as simplicial or cubical complexes, by reducing their number of cells. Consequently, they provide a bound for the Betti numbers (the most basic homological information). A discrete gradient vector field can eventually reduce the complex to its minimal form, having as many cells of each dimension as its corresponding Betti number, but this is not guaranteed. Moreover, finding an optimal discrete gradient vector field is an NP-hard problem. We describe here a generalization, which we call Homological Discrete Vector Field (HDVF), which can overcome these limitations by allowing cycles under a certain algebraic condition. In this work we define the HDVF and its associated reduction, we study how to efficiently compute a HDVF, we establish the relation between the HDVF and other concepts in computational homology and we estimate the average complexity of its computation. We also introduce five basic operations for modifying a HDVF, which can also be applied to discrete gradient vector fields.
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Article dans une revue
Topology and its Applications, Elsevier, 2017, 228, 〈10.1016/j.topol.2017.05.008〉
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https://hal.archives-ouvertes.fr/hal-01528103
Contributeur : Aldo Gonzalez-Lorenzo <>
Soumis le : samedi 27 mai 2017 - 11:42:49
Dernière modification le : mercredi 12 septembre 2018 - 01:26:04

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Aldo Gonzalez-Lorenzo, Alexandra Bac, Jean-Luc Mari, Pedro Real. Allowing cycles in discrete Morse theory. Topology and its Applications, Elsevier, 2017, 228, 〈10.1016/j.topol.2017.05.008〉. 〈hal-01528103〉

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