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Parameterized complexity of discrete Morse theory
 SCG ’13: Proceedings of the 29th Annual Symposium on Computational Geometry, ACM
"... Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3manifolds. However, such matchings are known to be NPhard to compute on 3manifolds, through a reduction to the erasability problem. Her ..."
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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 3manifolds. However, such matchings are known to be NPhard to compute on 3manifolds, 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 3manifolds which is fixedparameter tractable in the treewidth of the bipartite graph representing the adjacency of the 1 and 2simplexes. This algorithm also shows fixed parameter tractability for problems such as erasability and maximum alternating cyclefree matching. We further show that these results are also true when the treewidth of the dual graph of the triangulated 3manifold is bounded. Finally, we investigate the respective treewidths of simplicial and generalized triangulations of 3manifolds.
Characterizing Molecular Interactions in Chemical Systems
"... Fig. 1. Visual and quantitative exploration of covalent and noncovalent bonds in the βsheet polipeptide. Our analysis enables to visualize, enumerate, classify, and investigate molecular interactions in complex chemical systems. In this example, the amplitude of the signed electron density (ρ̃, c ..."
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Fig. 1. Visual and quantitative exploration of covalent and noncovalent bonds in the βsheet polipeptide. Our analysis enables to visualize, enumerate, classify, and investigate molecular interactions in complex chemical systems. In this example, the amplitude of the signed electron density (ρ̃, colorcoded from blue to red) enables to distinguish covalent bonds (yellow) from hydrogen bonds (cyan) and van der Waals interactions (dark blue). While the numerical integration of ∇ρ ̃ (right inset) enables to visually distinguish the latter two types of interactions, our combinatorial pipeline robustly extracts these features to support further quantitative analysis. In particular, our algorithm reveals the repeating pattern (black frame) of noncovalent interactions responsible for the folding of this molecule, which decomposes it in unitary building blocks corresponding to the elementary aminoacids composing the molecule. Abstract — Interactions between atoms have a major influence on the chemical properties of molecular systems. While covalent interactions impose the structural integrity of molecules, noncovalent interactions govern more subtle phenomena such as protein folding, bonding or self assembly. The understanding of these types of interactions is necessary for the interpretation of many biological processes and chemical design tasks. While traditionally the electron density is analyzed to interpret the quantum chemistry of a molecular system, noncovalent interactions are characterized by low electron densities and only slight variations of them – challenging their extraction and characterization. Recently, the signed electron density and the reduced gradient, two scalar fields derived from the electron density, have drawn much attention in quantum chemistry since they enable a qualitative visualization of these interactions even in complex molecular systems and experimental measurements. In this work, we present the first combinatorial algorithm for the
Conforming MorseSmale Complexes
"... of a scalar function (a) to generate an origin/destination map. (c) This map is used to compute an MS complex that conforms to the numericallycomputed geometry. (d,e) The ascending/descending manifolds of the MS complex form one such origin/destination map, but our algorithm allows for arbitrary ma ..."
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of a scalar function (a) to generate an origin/destination map. (c) This map is used to compute an MS complex that conforms to the numericallycomputed geometry. (d,e) The ascending/descending manifolds of the MS complex form one such origin/destination map, but our algorithm allows for arbitrary maps that enable a user to edit the MS complex. (f) Using the edited maps and using them as input for another MS complex computation, we can construct MS complexes that conform with any userspecified map. Abstract — MorseSmale (MS) complexes have been gaining popularity as a tool for featuredriven data analysis and visualization. However, the quality of their geometric embedding and the sole dependence on the input scalar field data can limit their applicability when expressing applicationdependent features. In this paper we introduce a new combinatorial technique to compute an MS complex that conforms to both an input scalar field and an additional, prior segmentation of the domain. The segmentation constrains the MS complex computation guaranteeing that boundaries in the segmentation are captured as separatrices of the MS complex. We demonstrate the utility and versatility of our approach with two applications. First, we use streamline integration to determine numerically computed basins/mountains and use the resulting segmentation as an input to our algorithm. This strategy enables the incorporation of prior flow path knowledge, effectively resulting in an MS complex that is as geometrically accurate as the employed numerical integration. Our second use case is motivated by the observation that often the data itself does not explicitly contain features known to be present by a domain expert. We introduce edit operations for MS complexes so that a user can directly modify their features while maintaining all the advantages of a robust topologybased representation.
Computing Persistent Homology via Discrete Morse Theory
, 2013
"... This report provides theoretical justification for the use of discrete Morse theory for the computation of homology and persistent homology, an overview of the state of the art for the computation of discrete Morse matchings and motivation for an interest in these computations, particularly from th ..."
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This report provides theoretical justification for the use of discrete Morse theory for the computation of homology and persistent homology, an overview of the state of the art for the computation of discrete Morse matchings and motivation for an interest in these computations, particularly from the point of view of topological data analysis. Additionally, a new simulated annealing based method for computing discrete Morse matchings is presented. For several problem instances this outperforms the best known heuristics for the task. The computation of homology and persistent homology has become an important task in computational topology, with applications in fields such as topological data analysis, computer vision and materials science. Unfortunately computing homology is currently infeasible for large input complexes. Discrete Morse theory enables the preprocessing of homology computation by reducing the size of the input complexes. This is advantageous from a