Optimal Morse matchings reveal essential structures of cell complexes that 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-simplices. 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 discuss the topological significance of the chosen parameters and investigate the respective treewidths of simplicial and generalized triangulations of 3-manifolds.
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.
Figure 1. Topology-aware denoising of a measured fluid velocity field: (left) original field, (middle) gaussian denoising, (right) gaussian denoising preserving topological singularities selected through our interface.Abstract-Recent developments in data acquisition technology enable to directly capture real vector fields, helping for a better understanding of physical phenomena. However measured data is corrupted by noise, puzzling the understanding of the phenomena. This turns the task of removing noise, i.e. denoising, an essential preprocessing step for a better analysis of the data. Nonetheless a careful use of denoising is required since usual algorithms not only remove the noise but can also eliminate information, in particular the vector field singularities, which are fundamental features in the analysis. This paper proposes a semi-automatic vector field denoising methodology, where the user visually controls the topological changes caused by classical vector field filtering in scale-spaces.
Figure 1. A simple discontinuous vector field (left) pertubed with a gaussian additive noise (middle left). The gaussian filter (middle right) blurs the interface, while the random walk (right) preserves it.Abstract-In recent years, several devices allow to directly measure real vector fields, leading to a better understanding of fundamental phenomena such as fluid simulation or brain water movement. This turns vector field visualization and analysis important tools for many applications in engineering and in medicine. However, real data is generally corrupted by noise, puzzling the understanding provided by those tools. Those tools thus need a denoising step as preprocessing, although usual denoising removes discontinuities, which are fundamental for vector field analysis. This paper proposes a novel method for vector field denoising based on random walks which preserve those discontinuities. It works in a meshless setting; it is fast, simple to implement, and shows a better performance than the traditional gaussian denoising technique.
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