Improved Bounds for Matroid Partition and Intersection Algorithms

We give bounds on total lengths of augmenting paths in standard implementations of the matroid partition and intersection algorithms, and indicate how these observations can be used to improve the running times in certain applications. For example, for the matroid intersection algorithm on two r by n matrices the running time is shown to be $O(nr^2 \log r)$. We also give improved versions of the two algorithms, when running times are measured in terms of calls to an independence oracle. For example, there is a matroid partition algorithm on $O(n)$n-element matroids using $O(n^{2.5} )$ independence tests.