Ra Rudi Pendavingh

On the number of matroids

We consider the problem of determining mn, the number of matroids on n elements. The best known lower bound on mn is due to Knuth (1974) who showed that loglogmn is at least n − 3/2logn − O(1). On the other hand, Piff (1973) showed that loglogmn ≤ n − logn + loglogn + O(1), and it has been conjectured since that the right answer is perhaps closer to Knuth’s bound.We show that this is indeed the case, and prove an upper bound on loglogmn that is within an additive 1+o(1) term of Knuth’s lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of stable sets in the Johnson graph to give a compressed representation of matroids.

Recognizing DNA graphs is difficult

DNA graphs are the vertex induced subgraphs of De Bruijn graphs over a four letter alphabet. In this paper, we prove the NP-hardness of various recognition problems for subgraphs of De Bruijn graphs; in particular, the recognition of DNA graphs is shown to be NP-hard. As a consequence, two open questions from a recent paper by Biazewicz et al. (Discrete Appl. Math. 98, (1999) 1) are answered in the negative.

Skew partial fields, multilinear representations of matroids, and a matrix tree theorem

We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tuttes definition, using chain groups. We show how such representations behave under duality and minors, we extend Tuttes representability criterion to this new class, and we study the generator matrices of the chain groups. An example shows that the class of matroids representable over a skew partial field properly contains the class of matroids representable over a skew field. Next, we show that every multilinear representation of a matroid can be seen as a representation over a skew partial field. Finally we study a class of matroids called quaternionic unimodular. We prove a generalization of the Matrix Tree theorem for this class.

A linear programming formulation of Mader's edge-disjoint paths problem

We give a dual pair of linear programs for a min-max result of Mader describing the maximum number of edge-disjoint T-paths in a graph G = (V, E) with T ⊆ V. We conclude that there exists a polynomial-time algorithm (based on the ellipsoid method) for finding the maximum number of T- paths in a capacitated graph, where the number of T-paths using an edge does not exceed the capacity of that edge.

The Magnus--Derek game revisited

We provide a new adaptive strategy for the maximizer in the Magnus-Derek game on the n-cycle. We show that this new strategy succeeds within O(nlogn) rounds, and thus improves on a predecessor result that uses a quadratic number of rounds.

Reconstructing a phylogenetic level-1 network from quartets.

We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on

Server Allocation Algorithms for Tiered Systems

n

On the number of bases of almost all matroids

n taxa from the set of all quartets containing a certain fixed taxon, in