622 research outputs found
Matroids with nine elements
We describe the computation of a catalogue containing all matroids with up to
nine elements, and present some fundamental data arising from this cataogue.
Our computation confirms and extends the results obtained in the 1960s by
Blackburn, Crapo and Higgs. The matroids and associated data are stored in an
online database, and we give three short examples of the use of this database.Comment: 22 page
Entropy of Closure Operators
The entropy of a closure operator has been recently proposed for the study of
network coding and secret sharing. In this paper, we study closure operators in
relation to their entropy. We first introduce four different kinds of rank
functions for a given closure operator, which determine bounds on the entropy
of that operator. This yields new axioms for matroids based on their closure
operators. We also determine necessary conditions for a large class of closure
operators to be solvable. We then define the Shannon entropy of a closure
operator, and use it to prove that the set of closure entropies is dense.
Finally, we justify why we focus on the solvability of closure operators only.Comment: arXiv admin note: substantial text overlap with arXiv:1209.655
Nonlinear Matroid Optimization and Experimental Design
We study the problem of optimizing nonlinear objective functions over
matroids presented by oracles or explicitly. Such functions can be interpreted
as the balancing of multi-criteria optimization. We provide a combinatorial
polynomial time algorithm for arbitrary oracle-presented matroids, that makes
repeated use of matroid intersection, and an algebraic algorithm for vectorial
matroids.
Our work is partly motivated by applications to minimum-aberration
model-fitting in experimental design in statistics, which we discuss and
demonstrate in detail
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