Guarantees for Greedy Maximization of Non-submodular Functions with Applications

We investigate the performance of the GREEDY algorithm for cardinality constrained maximization of non-submodular nondecreasing set functions. While there are strong theoretical guarantees on the performance of GREEDY for maximizing submodular functions, there are few guarantees for non-submodular ones. However, GREEDY enjoys strong empirical performance for many important non-submodular functions, e.g., the Bayesian A-optimality objective in experimental design. We prove theoretical guarantees supporting the empirical performance. Our guarantees are characterized by the (generalized) submodularity ratio γ and the (generalized) curvature α. In particular, we prove that GREEDY enjoys a tight approximation guarantee of 1 α (1− e−γα) for cardinality constrained maximization. In addition, we bound the submodularity ratio and curvature for several important real-world objectives, e.g., the Bayesian A-optimality objective, the determinantal function of a square submatrix and certain linear programs with combinatorial constraints. We experimentally validate our theoretical findings for several real-world applications.