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Deciding first order logic properties of matroids
[article]

2011
*
arXiv
*
pre-print

Here, we introduce an analogous notion for

arXiv:1108.5457v1
fatcat:sizqdul2nzechhgavbtpknxmby
*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 ...##
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Trees, grids, and MSO decidability: From graphs to matroids

2006
*
Theoretical Computer Science
*

(

doi:10.1016/j.tcs.2005.10.006
fatcat:dtsh4w7hs5dbtkz6nyqjef3cp4
*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. ...##
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Ordered matroids and regular independence systems

1996
*
Discrete Mathematics
*

We consider a class

doi:10.1016/0012-365x(94)00352-j
fatcat:t5gu3ytes5bvxnscyns6ebmuum
*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. ...##
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On Decidability of MSO Theories of Representable Matroids
[chapter]

2004
*
Lecture Notes in Computer Science
*

We show that, for every finite field , the class

doi:10.1007/978-3-540-28639-4_9
fatcat:36run6tgjrhtdpcu7ir6hdnxze
*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. ...##
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Amalgam width of matroids
[article]

2013
*
arXiv
*
pre-print

In particular, any

arXiv:1304.0299v2
fatcat:b7tgznfdfvbddkpboc2tadfyce
*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 ...##
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Amalgam Width of Matroids
[chapter]

2013
*
Lecture Notes in Computer Science
*

In particular, any

doi:10.1007/978-3-319-03898-8_23
fatcat:yod4q3b5tfgu5mfeblrxqu7rn4
*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] . ...##
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Positroids, non-crossing partitions, and positively oriented matroids

2014
*
Discrete Mathematics & Theoretical Computer Science
*

International audience We investigate the role that non-crossing partitions play in the study

doi:10.46298/dmtcs.2431
fatcat:vklnn5rudrbu5nse2ajo6mizjq
*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 ...##
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Monadic second-order model-checking on decomposable matroids
[article]

2011
*
arXiv
*
pre-print

We

arXiv:0908.4499v2
fatcat:zslper2a7bftbhigjrk4uv6zqy
*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. ...##
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Output-sensitive algorithm for generating the flats of a matroid
[article]

2011
*
arXiv
*
pre-print

In the case

arXiv:1107.4301v1
fatcat:cfrcmj2igng3vmnwh5gbzacrqm
*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. ...##
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First order convergence of matroids

2017
*
European journal of combinatorics (Print)
*

exists an infinite

doi:10.1016/j.ejc.2016.08.005
fatcat:i5kvbbg7vvfibmi4ytzsls55yu
*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. ...##
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First order convergence of matroids
[article]

2016
*
arXiv
*
pre-print

exists an infinite

arXiv:1501.06518v3
fatcat:ufrczsvzzbbs3lqvzzrndfeukm
*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. ...##
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Decomposition width of matroids

2012
*
Discrete Applied Mathematics
*

B 96 (2006), 325-351] showed that every

doi:10.1016/j.dam.2011.03.016
fatcat:rl7igzidhbeino3viz3xljlc2u
*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. ...##
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Decomposition Width of Matroids
[chapter]

2010
*
Lecture Notes in Computer Science
*

B 96 (2006), 325-351] showed that every

doi:10.1007/978-3-642-14165-2_6
fatcat:5kkgy24nrnbkjchzozeiysgtmi
*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. ...##
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Monadic second-order model-checking on decomposable matroids

2011
*
Discrete Applied Mathematics
*

We

doi:10.1016/j.dam.2011.02.005
fatcat:33f6zing5zg4jnuqy4arvae47a
*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. ...##
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Optimal assignments in an ordered set: An application of matroid theory

1968
*
Journal of Combinatorial Theory
*

Let X be a finite

doi:10.1016/s0021-9800(68)80039-0
fatcat:kpmo3vilkbcnrnuusbfxioo7cu
*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 ...
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