Submodular Maximization over Multiple Matroids via Generalized Exchange Properties

In this paper, we consider the problem of maximizing a non-negative submodular function f , defined on a (finite) ground set N , subject to matroid constraints. A function $f: 2^N \rightarrow {\mathbb R}$ is submodular if for all S , T *** N , f (S *** T ) + f (S *** T ) ≤ f (S ) + f (T ). Furthermore, all submodular functions that we deal with are assumed to be non-negative. Throughout, we assume that our submodular function f is given by a value oracle ; i.e., for a given set S *** N , an algorithm can query an oracle to find the value f (S ). Without loss of generality, we take the ground set N to be [n ] = {1,2,...,n }.

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