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Totally Free Expansions of Matroids
2002
Journal of combinatorial theory. Series B (Print)
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, over all totally free minors N of M, of the number of inequivalent F-representations of N. ...
A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. ...
ACKNOWLEDGMENTS Geelen was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada, Oxley and Vertigan were partially supported by grants from the National Security ...
doi:10.1006/jctb.2001.2068
fatcat:3qtpgglihrgdrepvnug3h2uqqu
A note on clone sets in representable matroids
2006
Discrete Mathematics
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We investigate the size of clone sets in representable matroids. Theorem 1.1. A 2-connected binary matroid has no clone sets. ...
If M is a 3-connected GF(3)-representable matroid that contains a clone set, then MU 2,4 . ...
A matroid M is totally free if and only if it is a totally free expansion of U 2,4 .
Lemma 2.3 (Geelen et al. [4, Proposition 4.3]). ...
doi:10.1016/j.disc.2006.05.005
fatcat:c33rfthhnvdydfvxl5ypl654qy
Page 7039 of Mathematical Reviews Vol. , Issue 2002J
[page]
2002
Mathematical Reviews
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free expansions of matroids. ...
(iii) The identification of all totally free quaternary and quinternary matroids. ...
On clone sets of GF(q)-representable matroids
2009
Discrete Mathematics
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This bound is given by investigating the representability of a class of near-uniform matroids. ...
We bound the size of a clone set in a 3-connected non-uniform GF(q)-representable matroid by a linear function of q. ...
We thank the anonymous referee for comments which improved the proofs and results of Section 4. ...
doi:10.1016/j.disc.2008.02.032
fatcat:rcjb6zfzsvgtpnkvo2luewvxla
Revenue submodularity
2009
Proceedings of the tenth ACM conference on Electronic commerce - EC '09
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We also give two applications of revenue submodularity: good approximation algorithms for novel market expansion problems, and approximate revenue guarantees for the VCG mechanism with i.i.d. bidders. ...
in matroid markets with i.i.d. bidders and "sufficient competition". ...
Thus, as in the proof of Theorem 3.2, the VCG mechanism inadvertently maximizes the total virtual value over the bases of the matroid. ...
doi:10.1145/1566374.1566409
dblp:conf/sigecom/DughmiRS09
fatcat:3k4icatlt5gq5cueaurizp7hd4
Revenue Submodularity
[chapter]
2009
Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering
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We also give two applications of revenue submodularity: good approximation algorithms for novel market expansion problems, and approximate revenue guarantees for the VCG mechanism with i.i.d. bidders. ...
in matroid markets with i.i.d. bidders and "sufficient competition". ...
Thus, as in the proof of Theorem 3.2, the VCG mechanism inadvertently maximizes the total virtual value over the bases of the matroid. ...
doi:10.1007/978-3-642-03821-1_13
fatcat:yur37nu7ezfvpjyag4vrrzrcpu
Revenue Submodularity
2012
Theory of Computing
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We also give two applications of revenue submodularity: good approximation algorithms for novel market expansion problems, and approximate revenue guarantees for the VCG mechanism with i.i.d. bidders. ...
in matroid markets with i.i.d. bidders and "sufficient competition." ...
Thus, as in the proof of Theorem 3.2, the VCG mechanism inadvertently maximizes the total virtual value over the bases of the matroid. ...
doi:10.4086/toc.2012.v008a005
dblp:journals/toc/DughmiRS12
fatcat:hahb7ynpxrc3xljqmatya6ak2u
Network theory and transversal matroids
1980
Discrete Applied Mathematics
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Finally, we consider this classification of the networks and give matroidal characterization of the hierarchy classes. 311 312 A. Recski, M. Id ...
Various matroidal models for the solvability of active linear networks are described in a uni&d way and compared with one another. ...
in the expansion of det N. ...
doi:10.1016/0166-218x(80)90027-x
fatcat:olfmcau25fawxgrku63ife4mry
A quasisymmetric function for matroids
2009
European journal of combinatorics (Print)
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A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant: ...
minimize the total weight; • behaves simply under matroid duality; • has a simple expansion in terms of P-partition enumerators; • is a valuation on decompositions of matroid base polytopes. ...
of a free commutative algebra, that is, a polynomial algebra. ...
doi:10.1016/j.ejc.2008.12.007
fatcat:2nnu5o6mafafbbxxiszazzxpsa
A quasisymmetric function for matroids
[article]
2020
arXiv
pre-print
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of how many bases minimize the total weight, (4) behaves simply under matroid duality, (5) has a simple expansion in terms of P-partition enumerators, and (6) is a valuation on decompositions of matroid ...
for integer weight vectors that give minimum total weight to a unique base of the matroid, (3) is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track ...
Let Mat be the free Z-module consisting of formal Z-linear combinations of basis elements [M ] indexed by isomorphism classes of matroids M . ...
arXiv:math/0606646v4
fatcat:3pvkdhqkzfbm5ny57jxb7ydzae
Page 55 of Mathematical Reviews Vol. , Issue 92a
[page]
1992
Mathematical Reviews
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In the authors’ terminology, a symmetric function is said to be multiplicity free if its Schur function expansion has all coefficients equal to 0, 1 or —1. ...
exists a k-term arithmetic progression A which totally belongs to one of the A;’s. ...
Page 7306 of Mathematical Reviews Vol. , Issue 2003j
[page]
2003
Mathematical Reviews
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This result follows from
05 COMBINATORICS
7306
an expansion of p(A; A) in terms of the free convex sets of an an- timatroid A [G. Gordon and E. W. ...
The authors study this anti- matroid characteristic polynomial for the class of chordal graphs. A graph is chordal if every cycle of length 4 or greater has a chord. ...
Ported Tutte Functions of Extensors and Oriented Matroids
[article]
2006
arXiv
pre-print
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The corank-nullity polynomial, basis expansions with activities, and a geometric lattice expansion generalize to ported Tutte functions of oriented matroids. ...
Solutions F can distinguish different orientations of the same matroid. ...
Ariel Caticha, Physics Department of the University at Albany, for his invitation to participate in his graduate Geometry of Physics course; Joanna Ellis-Monaghan, Michel Las Vergnas, Gary Gordon, Alan ...
arXiv:math/0605707v2
fatcat:th3pufbranahhlo73y74f3giaa
Investigating solvability and complexity of linear active networks by means of matroids
1979
IEEE Transactions on Circuits and Systems
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I/, in case of regularity is the free mutroid, i.e., M-(E$u 1,') contains no circuits. ...
Now a subset of columns of B' is linearly independent iff the columns correspond to a cutset-free subgraph of H [20] . ...
Professor at the Department of Mathematics, University of Denmark Lyngby. He has carried out research in the fields of circuit theory, graph theory, matroid theory, and computational complexity. ...
doi:10.1109/tcs.1979.1084645
fatcat:7gbkyyeshbfldiipdk6fcwpq24
The image of a tropical linear space
[article]
2018
arXiv
pre-print
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We show under mild hypotheses that tropim_A(L) is realizable if L is and apply the tropical image to construct the stable sum of two tropical linear spaces without a disjoint pair of bases. ...
Given a tropical linear space L ⊆T^n and a matrix A ∈T^m × n, the image AL of L under A is typically not a tropical linear space. ...
Transversal matroids are the induced matroids of free matroids. Corollary 4.7 also follows from the description of Stiefel tropical linear spaces as the tropical image of free modules. ...
arXiv:1808.02150v1
fatcat:f5p56cripfaqvirgazt5c5lnd4
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