Many important problems in discrete optimization require maximization of a monotonic submodular function subject to matroid constraints. For these problems, a simple greedy algorithm is guaranteed to obtain near-optimal solutions. In this article, we extend this classic result to a general class of adaptive optimization problems under partial observability, where each choice can depend on observations resulting from past choices. Specifically, we prove that a natural adaptive greedy algorithm provides a 1/(p + 1) approximation for the problem of maximizing an adaptive monotone submodular function subject to p matroid constraints, and more generally over arbitrary p-independence systems. We illustrate the usefulness of our result on a complex adaptive match-making application.

@techreport{golovin11matroid,
	author = {Daniel Golovin and Andreas Krause},
	institution = {arXiv},
	title = {Adaptive Submodular Optimization under Matroid Constraints},
	year = {2011}}