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We consider fast algorithms for monotone submodular maximization subject to amatroid constraint. We assume that the matroid is given as input in an explicitform, and the goal is to obtain the best possible running times for importantmatroids. We develop a new algorithm for a \emph{general matroid constraint}with a $1 - 1/e - \epsilon$ approximation that achieves a fast running timeprovided we have a fast data structure for maintaining a maximum weight base inthe matroid through a sequence of decrease weight operations. We construct suchdata structures for graphic matroids and partition matroids, and we obtain the\emph{first algorithms} for these classes of matroids that achieve anearly-optimal, $1 - 1/e - \epsilon$ approximation, using a nearly-linearnumber of function evaluations and arithmetic operations.

Towards Nearly-linear Time Algorithms for Submodular Maximization with a Matroid Constraint