New performance guarantees for the greedy maximization of submodular set functions

We present new tight performance guarantees for the greedy maximization of nondecreasing submodular set functions. Our main result first provides a performance guarantee in terms of the overlap of the optimal and greedy solutions. As a consequence we improve performance guarantees of Nemhauser, Wolsey and Fisher (1978) and Conforti and Cornu\'ejols (1984) for maximization over subsets, which are at least half the size of the problem domain. As a further application, we obtain a new tight performance guarantee in terms of the cardinality of the problem domain.