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Computing Representations of Matroids of Bounded Branch-Width
[chapter]

*
STACS 2007
*

For every k ≥ 1 and two finite fields F and F , we design a polynomial-time algorithm that given a

doi:10.1007/978-3-540-70918-3_20
dblp:conf/stacs/Kral07
fatcat:pp44buzibnbu7nc3u6lldhkzfi
*matroid*M*of**branch*-*width*at most k represented over F decides whether M is*representable*over F and if ... so, it*computes*a*representation**of*M over F . ... the spring*of*2002, Ondřej Pangrác for sharing his insights into*matroid*theory, and Till Tantau for his*computational*complexity remarks. ...##
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Decomposition Width of Matroids
[chapter]

2010
*
Lecture Notes in Computer Science
*

We also relate the decomposition

doi:10.1007/978-3-642-14165-2_6
fatcat:5kkgy24nrnbkjchzozeiysgtmi
*width*to*matroid**branch*-*width*and discuss implications*of*our results with respect to known algorithms. ... expressible in the monadic second-order logic can be*computed*in linear time for*matroids*given by a decomposition with*bounded**width*. ... The comments*of*the anonymous referees on the conference version*of*the paper that helped to improve and clarify the presentation a lot and are also greatly appreciated. ...##
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Decomposition width of matroids

2012
*
Discrete Applied Mathematics
*

We also relate the decomposition

doi:10.1016/j.dam.2011.03.016
fatcat:rl7igzidhbeino3viz3xljlc2u
*width*to*matroid**branch*-*width*and discuss implications*of*our results with respect to known algorithms. ... expressible in the monadic second-order logic can be*computed*in linear time for*matroids*given by a decomposition with*bounded**width*. ... The comments*of*the anonymous referees on the conference version*of*the paper that helped to improve and clarify the presentation a lot and are also greatly appreciated. ...##
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Amalgam Width of Matroids
[chapter]

2013
*
Lecture Notes in Computer Science
*

The parameter is linearly related to

doi:10.1007/978-3-319-03898-8_23
fatcat:yod4q3b5tfgu5mfeblrxqu7rn4
*branch*-*width*on finitely*representable**matroids*, while still allowing the algorithmic application on non-*representable**matroids*(which is not possible for*branch*-*width*... We also prove that the Tutte polynomial can be*computed*in polynomial time for*matroids*with*bounded*amalgam-*width*. ... Similarly,*matroids**of**bounded**branch*-*width*can be introduced in terms*of*an operation taking two*matroids**of**bounded**branch*-*width*and producing a larger*matroid**of**bounded**branch*-*width*by glueing them ...##
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Some Hard Problems on Matroid Spikes

2007
*
Theory of Computing Systems
*

However, one can observe a radical structural change
when getting to

doi:10.1007/s00224-007-1307-5
fatcat:m2m4ooibbne2leibtycu5xd34y
*matroid**representations*over infinite fields: For example [4], F-*representable**matroids**of**bounded**branch*-*width*are well-quasi-ordered ... a large class*of*problems is ef- ficiently solvable on*matroids**of**bounded**branch*-*width*represented over finite fields,*bounding*the*branch*-*width*does not seem to help with solving those prob- lems on ...##
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Trees, grids, and MSO decidability: From graphs to matroids

2006
*
Theoretical Computer Science
*

More exactly we prove, for every finite field F, that any class

doi:10.1016/j.tcs.2005.10.006
fatcat:dtsh4w7hs5dbtkz6nyqjef3cp4
*of*F-*representable**matroids*with a decidable MSO theory must have uniformly*bounded**branch*-*width*. ... Moreover, we show that*bounding*the*branch*-*width**of*all*matroids*in general is not sufficient to obtain a decidable MSO theory. ... Courcelle, the organizer*of*the workshop on "Logic and Graph Transformations" at the ICGT 2004 conference, for inviting us to present our research results there. ...##
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On Matroid Properties Definable in the MSO Logic
[chapter]

2003
*
Lecture Notes in Computer Science
*

It has been proved by the author that all

doi:10.1007/978-3-540-45138-9_41
fatcat:6dozjd5o4nhavcpwbrlxlvfsi4
*matroid*properties definable in the monadic second-order (MSO) logic can be recognized in polynomial time for*matroids**of**bounded**branch*-*width*which are represented ... In this work we review the MSO theory*of*finite*matroids*and show some interesting*matroid*properties which are MSO-definable. In particular, all minorclosed properties are recognizable in such way. ... Rod Downey for pointing my research towards parametrized complexity*of**matroid*problems. ...##
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Decomposition width - a new width parameter for matroids
[article]

2009
*
arXiv
*
pre-print

Since decompositions

arXiv:0904.2785v1
fatcat:jyyg4ed2praelecuxtzim7g5gu
*of*small*width*for our new notion can be*computed*in polynomial time for*matroids**of**bounded**branch*-*width*represented over finite fields, our results include recent algorithmic results ... We introduce a new*width*parameter for*matroids*called decomposition*width*and prove that every*matroid*property expressible in the monadic second order logic can be*computed*in linear time for*matroids*... Acknowledgement The author would like to thank Bruno Courcelle for his valuable comments on the conference paper [18] and Ondřej Pangrác for his insightful comments on the content*of*this paper as well ...##
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Amalgam width of matroids
[article]

2013
*
arXiv
*
pre-print

The parameter is linearly related to

arXiv:1304.0299v2
fatcat:b7tgznfdfvbddkpboc2tadfyce
*branch*-*width*on finitely*representable**matroids*(which is not possible for*branch*-*width*). ... We also prove that the Tutte polynomial can be*computed*in polynomial time for*matroids*with*bounded*amalgam*width*. ... Next, we show that the amalgam*width*is a generalization*of*the*branch*-*width*parameter for finitely*representable**matroids*in the sense that for such a class*of*matriods a*bounded*value*of**branch*-*width*...##
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On Decidability of MSO Theories of Representable Matroids
[chapter]

2004
*
Lecture Notes in Computer Science
*

In the other direction, we prove that every class

doi:10.1007/978-3-540-28639-4_9
fatcat:36run6tgjrhtdpcu7ir6hdnxze
*of*-*representable**matroids*with a decidable MSO theory must have uniformly*bounded**branch*-*width*. ... We show that, for every finite field , the class*of*allrepresentable*matroids**of**branch*-*width*at most a constant t has a decidable MSO theory. ... Nishimura, the organizers*of*the Dagstuhl Seminar in 2003 on "'Fixed Parameter Algorithms"' for inviting us to this interesting workshop, where the idea for this paper was born. ...##
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Deciding first order logic properties of matroids
[article]

2011
*
arXiv
*
pre-print

with locally

arXiv:1108.5457v1
fatcat:sizqdul2nzechhgavbtpknxmby
*bounded**branch*-*width*. ... Here, we introduce an analogous notion for*matroids*(locally*bounded**branch*-*width*) and show the existence*of*a fixed parameter algorithm for first order logic properties in classes*of*regular*matroids*... Acknowledgement The authors would like to thank Ken-ichi Kawarabayashi for discussing the details*of*his fixed parameter algorithm for*computing*multiway cuts given in [18] during the NII workshop-Graph ...##
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Finding Branch-Decompositions and Rank-Decompositions

2008
*
SIAM journal on computing (Print)
*

For that we use an algorithm that, for an input

doi:10.1137/070685920
fatcat:ybfpvrma5vbofhyzpqvcr4otrq
*matroid*represented over a fixed finite field, outputs its*branch*-decomposition*of**width*at most k if such exists. ... (The previous best algorithm for construction*of*a branchdecomposition or a rank-decomposition*of*optimal*width*due to Oum and Seymour [Testing*branch*-*width*. J. Combin. Theory Ser. ... Rank-*width*turns out to be very useful for algorithms on graphs*of**bounded*clique-*width*, since a class*of*graphs has*bounded*rank-*width*if and only if it has*bounded*clique-*width*. ...##
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A Parametrized Algorithm for Matroid Branch-Width

2005
*
SIAM journal on computing (Print)
*

We present an algorithm that, for a given

doi:10.1137/s0097539702418589
fatcat:xh63xswpzzgunopazww3npiybq
*matroid*M*of**bounded**branch*-*width*t which is represented over a finite field, finds a*branch*decomposition*of*M*of**width*at most 3t in cubic time. ... Other applications include recognition*of**matroid*properties definable in the monadic second-order logic for*bounded**branch*-*width*, or [Oum] a cubic approximation algorithm for graph rank-*width*and clique-*width*... Acknowledgments We would like to thank Geoff Whittle for valuable discussions about*matroid**representations*and*branch*-*width*, and the anonymous referee for helpful suggestions to improve the presentation ...##
###
On the Excluded Minors for Matroids of Branch-Width Three

2002
*
Electronic Journal of Combinatorics
*

Whittle, On

doi:10.37236/1648
fatcat:rzviyjy7cff2hnjnnzkutlkoti
*Matroids**of**Branch*-*Width*Three, submitted 2001] that if $M$ is an excluded minor for*matroids**of**branch*-*width*$3$, then $ M$ has at most $14$ elements. ... We also construct numbers*of*such ternary and quaternary*matroids*$ M$, and provide a simple practical algorithm for finding a*width*-$3$*branch*-decomposition*of*a*matroid*. ... Besides others, we want to mention the following recent works: well-quasi-ordering*of**matroids**of**bounded**branch*-*width*over finite fields [4] , sizebounds on the excluded minors for*matroids**of*fixed ...##
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Monadic second-order model-checking on decomposable matroids
[article]

2011
*
arXiv
*
pre-print

We first give a proof

arXiv:0908.4499v2
fatcat:zslper2a7bftbhigjrk4uv6zqy
*of*the polynomial time model-checking*of*monadic second-order formulas on*representable**matroids**of**bounded**branch*-*width*, by reduction to monadic second-order formulas on trees. ... We also provide a link between our logical approach and a grammar that allows to build*matroids**of**bounded**branch*-*width*. ... I am also very grateful to Brice Minaud and Pierre Clairambault for discussions we had respectively about*matroid**representations*and categories. ...
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