License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2021.32
URN: urn:nbn:de:0030-drops-146139
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14613/
Colin de Verdière, Éric ;
Magnard, Thomas
An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes
Abstract
We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface.
It is known that the problem admits an algorithm with running time f(c)n^{O(c)}, where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of the polynomial depends on C. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing a deterministic f(c)n³-time algorithm. We also provide a randomized algorithm with expected running time 2^{c^{O(1)}}n^{O(1)}.
Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface; the only elaborated tool that we use is an algorithm to compute grid minors.
BibTeX - Entry
@InProceedings{colindeverdiere_et_al:LIPIcs.ESA.2021.32,
author = {Colin de Verdi\`{e}re, \'{E}ric and Magnard, Thomas},
title = {{An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {32:1--32:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14613},
URN = {urn:nbn:de:0030-drops-146139},
doi = {10.4230/LIPIcs.ESA.2021.32},
annote = {Keywords: computational topology, embedding, simplicial complex, graph, surface, fixed-parameter tractability}
}
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Keywords: |
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computational topology, embedding, simplicial complex, graph, surface, fixed-parameter tractability |
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Collection: |
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29th Annual European Symposium on Algorithms (ESA 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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31.08.2021 |
