The complexity of detecting taut angle structures on triangulations

Burton, Benjamin A.; Spreer, Jonathan

doi:10.1137/1.9781611973105.13

Cited by 11 publications

(18 citation statements)

References 22 publications

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“…Hardness results are scarce in 3-dimensional computational topology, and to our knowledge all the other difficulty results are deduced from the Agol, Hass, and Thurston construction [1], except for the recent hardness results on computing taut angle structures [4] and optimal Morse matchings [5]. Our result displays a different intractability aspect of this theory.…”

Section: Introductionmentioning

confidence: 76%

On the complexity of immersed normal surfaces

Burton¹,

Verdière²,

Mesmay³

2016

Geom. Topol.

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Section: Introductionmentioning

confidence: 76%

On the complexity of immersed normal surfaces

Burton¹,

Verdière²,

Mesmay³

2016

Geom. Topol.

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“…To solve this problem, we propose an explicit algorithm for computing maximal alternating cycle-free matchings which is fixed-parameter tractable in the treewidth of this bipartite graph (Theorem 5). Furthermore, we show that finding optimal Morse matchings on triangulated 3-manifolds is also fixed-parameter tractable in the treewidth of the dual graph of the triangulation (Theorem 6), which is a common parameter when working with triangulated 3-manifolds [10].…”

Section: Introductionmentioning

confidence: 99%

Parameterized complexity of discrete morse theory

Burton¹,

Lewiner²,

Paixão³

et al. 2013

Proceedings of the 29th Annual Symposium on Symposuim on Computational Geometry - SoCG '13

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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 1and 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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“…In this way, we also show that the W -hierarchy as a purely complexity theoretical tool can be used in a very natural way to answer questions in the field of computational topology. Although there are many results about the computational complexity of topological problems [2,10,19,34,42], to the authors' knowledge, erasability is the first purely geometric problem shown to be W [P ]-complete.…”

Section: Introductionmentioning

confidence: 99%

“…To solve this problem, we propose an explicit algorithm for computing maximal alternating cycle-free matchings which is fixedparameter tractable in the treewidth of this bipartite graph (Theorem 5). Furthermore, we show that finding optimal Morse matchings on triangulated 3-manifolds is also fixedparameter tractable in the treewidth of the dual graph of the triangulation (Theorem 6), which is a common parameter when working with triangulated 3-manifolds [10].…”

Section: Introductionmentioning

confidence: 99%

Parameterized complexity of discrete morse theory

Burton

Lewiner

Paixão

et al. 2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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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.

show abstract

The complexity of detecting taut angle structures on triangulations

Cited by 11 publications

References 22 publications

On the complexity of immersed normal surfaces

On the complexity of immersed normal surfaces

Parameterized complexity of discrete morse theory

Parameterized complexity of discrete morse theory

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