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Constructive algorithm for path-width of matroids

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Date
2016
Link to item file
https://arxiv.org/abs/1507.02184v1
Dewey
Programmation, logiciels, organisation des données
Sujet
algorithms
DOI
http://dx.doi.org/10.1137/1.9781611974331.ch116
Conference name
27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2016)
Conference date
01-2016
Conference city
Arlington, Virginia
Conference country
United States
Book title
Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms
Author
Krauthgamer, Robert
Publisher
Society for Industrial and Applied Mathematics
Year
2016
Pages number
2106 (3 Vols)
ISBN
978-1-61197-433-1
Book URL
10.1137/1.9781611974331
URI
https://basepub.dauphine.fr/handle/123456789/19023
Collections
  • LAMSADE : Publications
Metadata
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Author
Jeong, Jisu
107170 Department of Mathematical Sciences, KAIST
Kim, Eun Jung
989 Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Oum, Sang-il
107170 Department of Mathematical Sciences, KAIST
Type
Communication / Conférence
Item number of pages
1695-1704
Abstract (EN)
Given n subspaces of a finite-dimensional 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 1-dimensional subspaces, this problem is equivalent to computing the path-width of an F-represented matroid in matroid theory and computing the trellis-width (or minimum trellis state-complexity) of a linear code in coding theory.We present a fixed-parameter tractable algorithm to construct a linear layout of width at most k, if it exists, for input subspaces of a finite-dimensional vector space over F. As corollaries, we obtain a fixed-parameter tractable algorithm to produce a path-decomposition of width at most k for an input F-represented matroid of path-width at most k, and a fixed-parameter tractable algorithm to find a linear rank-decomposition of width at most k for an input graph of linear rank-width 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 path-width and tree-width of graphs.It was previously known that a fixed-parameter tractable algorithm exists for the decision version of the problem for matroid path-width; a theorem by Geelen, Gerards, and Whittle (2002) implies that for each fixed finite field F, there are finitely many forbidden F-representable minors for the class of matroids of path-width at most k. An algorithm by Hliněný (2006) can detect a minor in an input F-represented matroid of bounded branch-width. However, this indirect approach would not produce an actual path-decomposition even if the complete list of forbidden minors were known. Our algorithm is the first one to construct such a path-decomposition and does not depend on the finiteness of forbidden minors.

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