Funk, D., Mayhew, D. orcid.org/0000-0003-4086-0980 and Newman, M. (2022) Tree Automata and Pigeonhole Classes of Matroids: I. Algorithmica, 84 (7). pp. 1795-1834. ISSN 0178-4617
Abstract
Hlinˇený’s Theorem shows that any sentence in the monadic second-order logic of matroids can be tested in polynomial time, when the input is limited to a class of F-representable matroids with bounded branch-width (where F is a finite field). If each matroid in a class can be decomposed by a subcubic tree in such a way that only a bounded amount of information flows across displayed separations, then the class has bounded decomposition-width. We introduce the pigeonhole property for classes of matroids: if every subclass with bounded branch-width also has bounded decomposition-width, then the class is pigeonhole. An efficiently pigeonhole class has a stronger property, involving an efficiently-computable equivalence relation on subsets of the ground set. We show that Hlinˇený’s Theorem extends to any efficiently pigeonhole class. In a sequel paper, we use these ideas to extend Hlinˇený’s Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, where H is any finite group. We also give a characterisation of the families of hypergraphs that can be described via tree automata: a family is defined by a tree automaton if and only if it has bounded decomposition-width. Furthermore, we show that if a class of matroids has the pigeonhole property, and can be defined in monadic second-order logic, then any subclass with bounded branch-width has a decidable monadic second-order theory.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © the Authors. This is an open access article under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited. |
Keywords: | Matroid theory; Tree automata; Monadic second-order logic |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 25 Jan 2024 11:26 |
Last Modified: | 25 Jan 2024 11:30 |
Published Version: | http://dx.doi.org/10.1007/s00453-022-00939-7 |
Status: | Published |
Publisher: | Springer |
Identification Number: | 10.1007/s00453-022-00939-7 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:208138 |