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

Daniel Král'
STACS 2007  
For every k ≥ 1 and two finite fields F and F , we design a polynomial-time algorithm that given a 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.  ... 
doi:10.1007/978-3-540-70918-3_20 dblp:conf/stacs/Kral07 fatcat:pp44buzibnbu7nc3u6lldhkzfi

Decomposition Width of Matroids [chapter]

Daniel Král'
2010 Lecture Notes in Computer Science  
We also relate the decomposition 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.  ... 
doi:10.1007/978-3-642-14165-2_6 fatcat:5kkgy24nrnbkjchzozeiysgtmi

Decomposition width of matroids

Daniel Král'
2012 Discrete Applied Mathematics  
We also relate the decomposition 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.  ... 
doi:10.1016/j.dam.2011.03.016 fatcat:rl7igzidhbeino3viz3xljlc2u

Amalgam Width of Matroids [chapter]

Lukáš Mach, Tomáš Toufar
2013 Lecture Notes in Computer Science  
The parameter is linearly related to 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  ... 
doi:10.1007/978-3-319-03898-8_23 fatcat:yod4q3b5tfgu5mfeblrxqu7rn4

Some Hard Problems on Matroid Spikes

Petr Hlineny
2007 Theory of Computing Systems  
However, one can observe a radical structural change when getting to 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  ... 
doi:10.1007/s00224-007-1307-5 fatcat:m2m4ooibbne2leibtycu5xd34y

Trees, grids, and MSO decidability: From graphs to matroids

Petr Hliněný, Detlef Seese
2006 Theoretical Computer Science  
More exactly we prove, for every finite field F, that any class 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.  ... 
doi:10.1016/j.tcs.2005.10.006 fatcat:dtsh4w7hs5dbtkz6nyqjef3cp4

On Matroid Properties Definable in the MSO Logic [chapter]

Petr Hliněný
2003 Lecture Notes in Computer Science  
It has been proved by the author that all 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.  ... 
doi:10.1007/978-3-540-45138-9_41 fatcat:6dozjd5o4nhavcpwbrlxlvfsi4

Decomposition width - a new width parameter for matroids [article]

Daniel Kral
2009 arXiv   pre-print
Since decompositions 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  ... 
arXiv:0904.2785v1 fatcat:jyyg4ed2praelecuxtzim7g5gu

Amalgam width of matroids [article]

Lukas Mach, Tomas Toufar
2013 arXiv   pre-print
The parameter is linearly related to 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  ... 
arXiv:1304.0299v2 fatcat:b7tgznfdfvbddkpboc2tadfyce

On Decidability of MSO Theories of Representable Matroids [chapter]

Petr Hliněný, Detlef Seese
2004 Lecture Notes in Computer Science  
In the other direction, we prove that every class 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.  ... 
doi:10.1007/978-3-540-28639-4_9 fatcat:36run6tgjrhtdpcu7ir6hdnxze

Deciding first order logic properties of matroids [article]

Tomas Gavenciak and Daniel Kral and Sang-il Oum
2011 arXiv   pre-print
with locally 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  ... 
arXiv:1108.5457v1 fatcat:sizqdul2nzechhgavbtpknxmby

Finding Branch-Decompositions and Rank-Decompositions

Petr Hliněný, Sang-il Oum
2008 SIAM journal on computing (Print)  
For that we use an algorithm that, for an input 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.  ... 
doi:10.1137/070685920 fatcat:ybfpvrma5vbofhyzpqvcr4otrq

A Parametrized Algorithm for Matroid Branch-Width

Petr Hlinený
2005 SIAM journal on computing (Print)  
We present an algorithm that, for a given 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  ... 
doi:10.1137/s0097539702418589 fatcat:xh63xswpzzgunopazww3npiybq

On the Excluded Minors for Matroids of Branch-Width Three

Petr Hliněný
2002 Electronic Journal of Combinatorics  
Whittle, On 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  ... 
doi:10.37236/1648 fatcat:rzviyjy7cff2hnjnnzkutlkoti

Monadic second-order model-checking on decomposable matroids [article]

Yann Strozecki
2011 arXiv   pre-print
We first give a proof 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.  ... 
arXiv:0908.4499v2 fatcat:zslper2a7bftbhigjrk4uv6zqy
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