Constructive algorithm for pathwidth of matroids
Jeong, Jisu; Kim, Eun Jung; Oum, Sangil (2016), Constructive algorithm for pathwidth of matroids, in Krauthgamer, Robert, Proceedings of the TwentySeventh Annual ACMSIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, p. 16951704. 10.1137/1.9781611974331.ch116
Type
Communication / ConférenceExternal document link
https://arxiv.org/abs/1507.02184v1Date
2016Conference title
27th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2016)Conference date
201601Conference city
Arlington, VirginiaConference country
United StatesBook title
Proceedings of the TwentySeventh Annual ACMSIAM Symposium on Discrete AlgorithmsBook author
Krauthgamer, RobertPublisher
Society for Industrial and Applied Mathematics
ISBN
9781611974331
Number of pages
2106 (3 Vols)Pages
16951704
Publication identifier
Metadata
Show full item recordAuthor(s)
Jeong, JisuDepartment of Mathematical Sciences, KAIST
Kim, Eun Jung
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Oum, Sangil
Department of Mathematical Sciences, KAIST
Abstract (EN)
Given n subspaces of a finitedimensional vector space over a fixed finite field F, we wish to find a linear layout V1, V2, …, Vn of the subspaces such that dim((V1 + V2 + ⃛ + Vi)∩(Vi+1 + ⃛ + Vn)) ≤ k for all i; such a linear layout is said to have width at most k. When restricted to 1dimensional subspaces, this problem is equivalent to computing the pathwidth of an Frepresented matroid in matroid theory and computing the trelliswidth (or minimum trellis statecomplexity) of a linear code in coding theory.We present a fixedparameter tractable algorithm to construct a linear layout of width at most k, if it exists, for input subspaces of a finitedimensional vector space over F. As corollaries, we obtain a fixedparameter tractable algorithm to produce a pathdecomposition of width at most k for an input Frepresented matroid of pathwidth at most k, and a fixedparameter tractable algorithm to find a linear rankdecomposition of width at most k for an input graph of linear rankwidth at most k. In both corollaries, no such algorithms were known previously. Our approach is based on dynamic programming combined with the idea developed by Bodlaender and Kloks (1996) for their work on pathwidth and treewidth of graphs.It was previously known that a fixedparameter tractable algorithm exists for the decision version of the problem for matroid pathwidth; a theorem by Geelen, Gerards, and Whittle (2002) implies that for each fixed finite field F, there are finitely many forbidden Frepresentable minors for the class of matroids of pathwidth at most k. An algorithm by Hliněný (2006) can detect a minor in an input Frepresented matroid of bounded branchwidth. However, this indirect approach would not produce an actual pathdecomposition even if the complete list of forbidden minors were known. Our algorithm is the first one to construct such a pathdecomposition and does not depend on the finiteness of forbidden minors.Subjects / Keywords
algorithmsRelated items
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