research-article

Parameterized complexity of discrete morse theory

Authors:

Benjamin A. Burton,

Thomas Lewiner,

Jonathan SpreerAuthors Info & Claims

SoCG '13: Proceedings of the twenty-ninth annual symposium on Computational geometry

Pages 127 - 136

https://doi.org/10.1145/2462356.2462391

Published: 17 June 2013 Publication History

Abstract

Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds, through a reduction to the erasability problem. Here, we refine the study of the complexity of problems related to discrete Morse theory in terms of parameterized complexity. On the one hand we prove that the erasability problem is W[P]-complete on the natural parameter. On the other hand we propose an algorithm for computing optimal Morse matchings on triangulations of 3-manifolds which is fixed-parameter tractable in the treewidth of the bipartite graph representing the adjacency of the 1- and 2-simplexes. This algorithm also shows fixed parameter tractability for problems such as erasability and maximum alternating cycle-free matching.

We further show that these results are also true when the treewidth of the dual graph of the triangulated 3-manifold is bounded. Finally, we investigate the respective treewidths of simplicial and generalized triangulations of 3-manifolds.

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Cited By

Chintakunta HGentimis TGonzalez-Diaz RJimenez MKrim H(2018)An entropy-based persistence barcodePattern Recognition10.1016/j.patcog.2014.06.02348:2(391-401)Online publication date: 30-Dec-2018
https://dl.acm.org/doi/10.1016/j.patcog.2014.06.023
Жукова АZhukova AПанина ГPanina G(2017)Дискретная теория Морса для пространств модулей шарнирных многоугольников, или пасьянс на окружностиDiscrete Morse theory for moduli spaces of flexible polygons, or solitaire game on the circleМатематический сборникMatematicheskii Sbornik10.4213/sm8677208:9(100-115)Online publication date: 2017
https://doi.org/10.4213/sm8677
Zhukova APanina G(2017)Discrete Morse theory for the moduli spaces of polygonal linkages, or solitaire on a circleSbornik: Mathematics10.1070/SM8677208:9(1353-1367)Online publication date: 28-Nov-2017
https://doi.org/10.1070/SM8677
Show More Cited By

Index Terms

Parameterized complexity of discrete morse theory
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SoCG '13: Proceedings of the twenty-ninth annual symposium on Computational geometry

June 2013

472 pages

ISBN:9781450320313

DOI:10.1145/2462356

General Chairs:
Guilherme D. da Fonseca
UNIRIO
,
Thomas Lewiner
PUC-Rio
,
Luis Peñaranda
IMPA
,
Program Chairs:
Timothy Chan
University of Waterloo
,
Rolf Klein
University of Bonn

Copyright © 2013 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 17 June 2013

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SoCG '13

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SoCG '13: Symposium on Computational Geometry 2013

June 17 - 20, 2013

Rio de Janeiro, Brazil

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SoCG '13 Paper Acceptance Rate 48 of 137 submissions, 35%;

Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Chintakunta HGentimis TGonzalez-Diaz RJimenez MKrim H(2018)An entropy-based persistence barcodePattern Recognition10.1016/j.patcog.2014.06.02348:2(391-401)Online publication date: 30-Dec-2018
https://dl.acm.org/doi/10.1016/j.patcog.2014.06.023
Жукова АZhukova AПанина ГPanina G(2017)Дискретная теория Морса для пространств модулей шарнирных многоугольников, или пасьянс на окружностиDiscrete Morse theory for moduli spaces of flexible polygons, or solitaire game on the circleМатематический сборникMatematicheskii Sbornik10.4213/sm8677208:9(100-115)Online publication date: 2017
https://doi.org/10.4213/sm8677
Zhukova APanina G(2017)Discrete Morse theory for the moduli spaces of polygonal linkages, or solitaire on a circleSbornik: Mathematics10.1070/SM8677208:9(1353-1367)Online publication date: 28-Nov-2017
https://doi.org/10.1070/SM8677
Rathod ABin Masood TNatarajan V(2017)Approximation algorithms for Max Morse MatchingComputational Geometry: Theory and Applications10.1016/j.comgeo.2016.10.00261:C(1-23)Online publication date: 1-Feb-2017
https://dl.acm.org/doi/10.1016/j.comgeo.2016.10.002
Burton BColin de Verdière Éde Mesmay A(2016)On the complexity of immersed normal surfacesGeometry & Topology10.2140/gt.2016.20.106120:2(1061-1083)Online publication date: 28-Apr-2016
https://doi.org/10.2140/gt.2016.20.1061
De Floriani LFugacci UIuricich FMagillo P(2015)Morse complexes for shape segmentation and homological analysisComputer Graphics Forum10.1111/cgf.1259634:2(761-785)Online publication date: 1-May-2015
https://dl.acm.org/doi/10.1111/cgf.12596
Burton BPettersson W(2014)Fixed Parameter Tractable Algorithms in Combinatorial TopologyComputing and Combinatorics10.1007/978-3-319-08783-2_26(300-311)Online publication date: 2014
https://doi.org/10.1007/978-3-319-08783-2_26

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