  We study the matroid secretary problems with submodular valuation functions.
In these problems, the elements arrive in random order. When one element
arrives, we have to make an immediate and irrevocable decision on whether to
accept it or not. The set of accepted elements must form an {\em independent
set} in a predefined matroid. Our objective is to maximize the value of the
accepted elements. In this paper, we focus on the case that the valuation
function is a non-negative and monotonically non-decreasing submodular
function.
  We introduce a general algorithm for such {\em submodular matroid secretary
problems}. In particular, we obtain constant competitive algorithms for the
cases of laminar matroids and transversal matroids. Our algorithms can be
further applied to any independent set system defined by the intersection of a
{\em constant} number of laminar matroids, while still achieving constant
competitive ratios. Notice that laminar matroids generalize uniform matroids
and partition matroids.
  On the other hand, when the underlying valuation function is linear, our
algorithm achieves a competitive ratio of 9.6 for laminar matroids, which
significantly improves the previous result.
