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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

Ordered matroids and regular independence systems

J.Orestes Cerdeira, Paulo Barcia
1996 Discrete Mathematics  
We consider a class of matroids which we call ordered matroids. We show that these are the matroids of regular independence systems.  ...  (If E is a finite ordered set, a regular independence system on E is an independence system (E, F) with the following property: if A E 9 and a E A, then (A -{a}) U {e} E 9 for all e E E-A such that e <  ...  Acknowledgements We are grateful to Ulrich Faigle (Twente University) for having brought to our knowledge the work on matroids from submodular functions on intersecting ring families.  ... 
doi:10.1016/0012-365x(94)00352-j fatcat:t5gu3ytes5bvxnscyns6ebmuum

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

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.  ...  MSO properties In this section, we show that the problem of deciding monadic second order properties becomes computationally tractable when we restrict ourselves to matroids of bounded amalgam width: Theorem  ... 
arXiv:1304.0299v2 fatcat:b7tgznfdfvbddkpboc2tadfyce

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

Positroids, non-crossing partitions, and positively oriented matroids

Federico Ardila, Felipe Rincón, Lauren Williams
2014 Discrete Mathematics & Theoretical Computer Science  
International audience We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov.  ...  We also prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space.  ...  Sturmfels proved [Stu87] that the existence of an algorithm for deciding if any given (oriented) matroid is realizable over Q is equivalent to the existence of an algorithm for deciding the solvability  ... 
doi:10.46298/dmtcs.2431 fatcat:vklnn5rudrbu5nse2ajo6mizjq

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.  ...  Finally, we introduce a new class of non-necessarily representable matroids, described by a grammar and on which monadic second-order formulas can be checked in linear time.  ...  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

Output-sensitive algorithm for generating the flats of a matroid [article]

A. Montina
2011 arXiv   pre-print
In the case of vectorial matroids, a specific algorithm is reported whose time complexity is equal to O(N^2 M d^2), d being the rank of the matroid.  ...  We present an output-sensitive algorithm for generating the whole set of flats of a finite matroid.  ...  Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.  ... 
arXiv:1107.4301v1 fatcat:cfrcmj2igng3vmnwh5gbzacrqm

First order convergence of matroids

František Kardoš, Daniel Král', Anita Liebenau, Lukáš Mach
2017 European journal of combinatorics (Print)  
exists an infinite matroid with the elements forming a probability space that has asymptotically the same first order properties.  ...  We establish the matroid counterpart of this result: every first order convergent sequence of matroids with bounded branch-depth representable over a fixed finite field has a limit modeling, i.e., there  ...  properties of first order convergent sequences.  ... 
doi:10.1016/j.ejc.2016.08.005 fatcat:i5kvbbg7vvfibmi4ytzsls55yu

First order convergence of matroids [article]

Frantisek Kardos and Daniel Kral and Anita Liebenau and Lukas Mach
2016 arXiv   pre-print
exists an infinite matroid with the elements forming a probability space that has asymptotically the same first order properties.  ...  We establish the matroid counterpart of this result: every first order convergent sequence of matroids with bounded branch-depth representable over a fixed finite field has a limit modeling, i.e., there  ...  properties of first order convergent sequences.  ... 
arXiv:1501.06518v3 fatcat:ufrczsvzzbbs3lqvzzrndfeukm

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  ...  We also relate the decomposition width to matroid branch-width and discuss implications of our results with respect to known algorithms.  ...  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  ...  We also relate the decomposition width to matroid branch-width and discuss implications of our results with respect to known algorithms.  ...  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

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.  ...  Finally, we introduce a new class of non-necessarily representable matroids, described by a grammar and on which monadic second-order formulas can be checked in linear time.  ...  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

Optimal assignments in an ordered set: An application of matroid theory

David Gale
1968 Journal of Combinatorial Theory  
Let X be a finite ordered set and let q~ be a function from X to subsets of a set Y. A subset A of X is called assignable if there is an injection ~b from A to Y that ~(a) ~ q~(a) for all a in A.  ...  It is shown that the assignable sets form a matroid on X and using this it is shown that there exists an optimal assignable set A, meaning that if A is any other assignable set then there is an injection  ...  Fulkerson to the concept of matroid and realized that what I had observed was a rather obvious property of these objects, so that for those familiar with the matroid literature my theorem could be proved  ... 
doi:10.1016/s0021-9800(68)80039-0 fatcat:kpmo3vilkbcnrnuusbfxioo7cu
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