214 research outputs found
Counting matroids in minor-closed classes
A flat cover is a collection of flats identifying the non-bases of a matroid.
We introduce the notion of cover complexity, the minimal size of such a flat
cover, as a measure for the complexity of a matroid, and present bounds on the
number of matroids on elements whose cover complexity is bounded. We apply
cover complexity to show that the class of matroids without an -minor is
asymptotically small in case is one of the sparse paving matroids
, , , , or , thus confirming a few special
cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other
hand, we show a lower bound on the number of matroids without -minor
which asymptoticaly matches the best known lower bound on the number of all
matroids, due to Knuth.Comment: 13 pages, 3 figure
Duality, matroids, qubits, twistors and surreal numbers
We show that via the Grassmann-Pl\"ucker relations, the various apparent
unrelated concepts, such as duality, matroids, qubits, twistors and surreal
numbers are, in fact, deeply connected. Moreover, we conjecture the possibility
that these concepts may be considered as underlying mathematical structures in
quantum gravity.Comment: 17 pages, published in Frontier in Physic
The College Admissions problem with lower and common quotas
We study two generalised stable matching problems motivated by the current matching scheme used in the higher education sector in Hungary. The first problem is an extension of the College Admissions problem in which the colleges have lower quotas as well as the normal upper quotas. Here, we show that a stable matching may not exist and we prove that the problem of determining whether one does is NP-complete in general. The second problem is a different extension in which, as usual, individual colleges have upper quotas, but, in addition, certain bounded subsets of colleges have common quotas smaller than the sum of their individual quotas. Again, we show that a stable matching may not exist and the related decision problem is NP-complete. On the other hand, we prove that, when the bounded sets form a nested set system, a stable matching can be found by generalising, in non-trivial ways, both the applicant-oriented and college-oriented versions of the classical Gale–Shapley algorithm. Finally, we present an alternative view of this nested case using the concept of choice functions, and with the aid of a matroid model we establish some interesting structural results for this case
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