We present a hierarchy of covering properties of rational convex cones with respect to the unimodular subcones spanned by the Hilbert basis.For two of the concepts from the hierarchy we derive characterizations: a description of partitions that leads to a natural integer programming formulation for the HILBERT PAR-TITION problem, and a characterization of "binary covers" that admits a linear algebra test over GF(2) for the existence of BINARY HILBERT COVERS.Implementation of our test leads to interesting new examples, among them: cones that have a HILBERT PARTITION but no REGULAR one; a four-dimensional cone with unimodular facets that has no HILBERT PARTITION; and two five-dimensional cones that do not have any BINARY HILBERT COVER.

This paper deals with the problem of representing the matching independence system in a graph as the intersection of finitely many matroids. After characterizing the graphs for which the matching independence system is the intersection of two matroids, we study the function µ(G), which is the minimum number of matroids that need to be intersected in order to obtain the set of matchings on a graph G, and examine the maximal value, µ(n), for graphs with n vertices. We describe an integer programming formulation for deciding whether µ(G) ≤ k. Using combinatorial arguments, we prove that µ(n) ∈ Ω(log log n). On the other hand, we establish that µ(n) ∈ O(log n/ log log n). Finally, we prove that µ(n) = 4 for n = 5, . . . , 12, and sketch a proof of µ(n)=5 for n = 13, 14, 15.

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This paper deals with the problem of representing the matching independence system in a graph as the intersection of finitely many matroids. After characterizing the graphs for which the matching independence system is the intersection of two matroids, we study the function µ(G), which is the minimum number of matroids that need to be intersected in order to obtain the set of matchings on a graph G, and examine the maximal value, µ(n), for graphs with n vertices. We describe an integer programming formulation for deciding whether µ(G) ≤ k. Using combinatorial arguments, we prove that µ(n) ∈ Ω(log log n). On the other hand, we establish that µ(n) ∈ O(log n/ log log n). Finally, we prove that µ(n) = 4 for n = 5, . . . , 12, and sketch a proof of µ(n)=5 for n = 13, 14, 15.

An algorithmic characterization of a particular combinatorial optimization problem means that there is an algorithm that works exact if and only if applied to the combinatorial optimization problem under investigation. According to Jack Edmonds, the Greedy algorithm leads to an algorithmic characterization of matroids. We deal here with the algorithmic characterization of the intersection of two matroids. To this end we introduce two different augmentation digraphs for the intersection of any two independence systems. Paths and cycles in these digraphs correspond to candidates for improving feasible solutions. The first digraph gives rise to an algorithmic characterization of bipartite b-matching. The second digraph leads to a polynomial-time augmentation algorithm for the (weighted) matroid intersection problem and to a conjecture about an algorithmic characterization of matroid intersection.Supported by a "Gerhard-Hess-Forschungsförderpreis" (WE 1462/2-2) of the German Science Foundation (DFG) awarded to R. Weismantel.