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Deciding first order logic properties of matroids [article]

Tomas Gavenciak and Daniel Kral and Sang-il Oum
2011 arXiv   pre-print
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  ...  ACM 48 (2006), 1184-1206] introduced a notion of graph classes with locally bounded tree-width and established that every first order logic property can be decided in almost linear time in such a graph  ...  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

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

Petr Hliněný, Detlef Seese
2006 Theoretical Computer Science  
(Matroids can be viewed as a wide generalization of graphs, and they seem to capture some second order properties in a more suitable way than graphs themselves, cf. the recent development in matroid structure  ...  Monadic second order (MSO) logic has proved to be a useful tool in many areas of application, reaching from decidability and complexity to picture processing, correctness of programs and parallel processes  ...  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 Decidability of MSO Theories of Representable Matroids [chapter]

Petr Hliněný, Detlef Seese
2004 Lecture Notes in Computer Science  
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.  ...  In the other direction, we prove that every class of -representable matroids with a decidable MSO theory must have uniformly bounded branch-width.  ...  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

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

Decomposition width of matroids

Daniel Král'
2012 Discrete Applied Mathematics  
B 96 (2006), 325-351] showed that every matroid property expressible in the monadic second-order logic can be decided in linear time for matroids with bounded branch-width that are represented over finite  ...  Hliněný, Branch-width, parse trees, and monadic second-order logic for matroids, J. Combin. Theory Ser.  ...  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

Decomposition Width of Matroids [chapter]

Daniel Král'
2010 Lecture Notes in Computer Science  
B 96 (2006), 325-351] showed that every matroid property expressible in the monadic second-order logic can be decided in linear time for matroids with bounded branch-width that are represented over finite  ...  Hliněný, Branch-width, parse trees, and monadic second-order logic for matroids, J. Combin. Theory Ser.  ...  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

Amalgam width of matroids [article]

Lukas Mach, Tomas Toufar
2013 arXiv   pre-print
In particular, any property expressible in the monadic second order logic can be decided in linear time for matroids with bounded amalgam-width.  ...  We introduce a new matroid width parameter based on the operation of matroid amalgamation, which we call amalgam-width.  ...  Research of this phenomenon culminated in proving a general theorem of Courcelle [2] , which asserts that any graph property expressible in the monadic second order (MSO) logic can be decided in linear  ... 
arXiv:1304.0299v2 fatcat:b7tgznfdfvbddkpboc2tadfyce

Branch-Width, Parse Trees, and Monadic Second-Order Logic for Matroids [chapter]

Petr Hliněný
2003 Lecture Notes in Computer Science  
We prove that if M is a family of matroids described by a sentence in the monadic second-order logic of matroids, then there is a finite tree automaton accepting exactly those parse trees which build vector  ...  We introduce "matroid parse trees" which, using only a limited amount of information at each node, can build up the vector representations of matroids of bounded branch-width over a finite field.  ...  Applying the language of MSO logic to such matroid structures gives the monadic second-order theory M S M of matroids: Definition.  ... 
doi:10.1007/3-540-36494-3_29 fatcat:wsejm3qjovbjbbt6nfeetf4ypm

Branch-width, parse trees, and monadic second-order logic for matroids

Petr Hliněný
2006 Journal of combinatorial theory. Series B (Print)  
We prove that if M is a family of matroids described by a sentence in the monadic second-order logic of matroids, then there is a finite tree automaton accepting exactly those parse trees which build vector  ...  We introduce "matroid parse trees" which, using only a limited amount of information at each node, can build up the vector representations of matroids of bounded branch-width over a finite field.  ...  Applying the language of MSO logic to such matroid structures gives the monadic second-order theory M S M of matroids: Definition.  ... 
doi:10.1016/j.jctb.2005.08.005 fatcat:rhduqrqesjepdk7elikt2beiku

Amalgam Width of Matroids [chapter]

Lukáš Mach, Tomáš Toufar
2013 Lecture Notes in Computer Science  
In particular, any property expressible in the monadic second order logic can be decided in linear time for matroids with bounded amalgam-width.  ...  We introduce a new matroid width parameter based on the operation of matroid amalgamation, which we call amalgam-width.  ...  Besides MSO properties, first order properties for matroids have been studied from algorithmic point of view in [11] .  ... 
doi:10.1007/978-3-319-03898-8_23 fatcat:yod4q3b5tfgu5mfeblrxqu7rn4

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

Daniel Kral
2009 arXiv   pre-print
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  ...  of Hlineny [J.  ...  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

Monadic second-order model-checking on decomposable matroids

Yann Strozecki
2011 Discrete Applied Mathematics  
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 one to build matroids of bounded branch-width.  ...  I am also very grateful to Brice Minaud and Pierre Clairambault for early discussions on matroids and pushout and to my advisor Arnaud Durand for his support and help while writing this article.  ... 
doi:10.1016/j.dam.2011.02.005 fatcat:33f6zing5zg4jnuqy4arvae47a

Some Hard Problems on Matroid Spikes

Petr Hlineny
2007 Theory of Computing Systems  
xj − 1 j∈[1,n], fj ∈X c) Otherwise, X is not a basis of S. 3 Easy and Hard MSO Properties Monadic second-order (MSO) logic is famous for its well-balanced  ...  Namely, the question whether a given spike is the free spike is co-N P -hard (though the property itself is definable in monadic second-order logic); and the task to compute the  ... 
doi:10.1007/s00224-007-1307-5 fatcat:m2m4ooibbne2leibtycu5xd34y

Width Parameters Beyond Tree-width and their Applications

P. Hlineny, S.-i. Oum, D. Seese, G. Gottlob
2007 Computer journal  
In this contribution, we demonstrate how 'width' parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the  ...  Besides the very successful concept of tree-width (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth.  ...  Dermietzel for his help with generating some of the figures. Our last, special thanks we reserve for the editors of this volume for the invitation and their constant encouragement.  ... 
doi:10.1093/comjnl/bxm052 fatcat:n4nbasuvubc5lkajkskqk22fey

Vertex-minors, monadic second-order logic, and a conjecture by Seese

Bruno Courcelle, Sang-il Oum
2007 Journal of combinatorial theory. Series B (Print)  
We obtain a slight weakening of a conjecture by Seese stating that sets of graphs having a decidable satisfiability problem for monadic second-order logic have bounded clique-width.  ...  We prove that one can express the vertex-minor relation on finite undirected graphs by formulas of monadic second-order logic (with no edge set quantification) extended with a predicate expressing that  ...  Section 5 reviews monadic second-order logic and its use for expressing properties and transformations of graphs, matroids, and isotropic systems.  ... 
doi:10.1016/j.jctb.2006.04.003 fatcat:3dfb5tqexndablnknwykkbcdlu
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