eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-28
23:1
23:17
10.4230/LIPIcs.ICALP.2022.23
article
Hodge Decomposition and General Laplacian Solvers for Embedded Simplicial Complexes
Black, Mitchell
1
Nayyeri, Amir
1
School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA
We describe a nearly-linear time algorithm to solve the linear system L₁x = b parameterized by the first Betti number of the complex, where L₁ is the 1-Laplacian of a simplicial complex K that is a subcomplex of a collapsible complex X linearly embedded in ℝ³. Our algorithm generalizes the work of Black et al. [SODA2022] that solved the same problem but required that K have trivial first homology. Our algorithm works for complexes K with arbitrary first homology with running time that is nearly-linear with respect to the size of the complex and polynomial with respect to the first Betti number. The key to our solver is a new algorithm for computing the Hodge decomposition of 1-chains of K in nearly-linear time. Additionally, our algorithm implies a nearly quadratic solver and nearly quadratic Hodge decomposition for the 1-Laplacian of any simplicial complex K embedded in ℝ³, as K can always be expanded to a collapsible embedded complex of quadratic complexity.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol229-icalp2022/LIPIcs.ICALP.2022.23/LIPIcs.ICALP.2022.23.pdf
Computational Topology
Laplacian solvers
Combinatorial Laplacian
Hodge decomposition
Parameterized Complexity