Near Optimal algorithms for constrained submodular programs with discounted cooperative costs

In this paper, we investigate two problems, namely (a) minimizing a submodular cost function under combinatorial constraints, which include cuts, matchings, paths, etc., and (b) optimizing a submodular function under submodular cover and submodular knapsack constraints. While both problems have hardness factors of Ω( √ n) for general submodular cost functions, we show how we can achieve constant approximation factors (in certain cases, even fully polynomial time approximation schemes), when we restrict the cost functions to low rank sums of concave over modular functions. A wide variety of machine learning applications are very naturally modeled via this subclass of submodular functions. Our work therefore provides a tighter connection between theory and practice by enabling theoretically satisfying guarantees for a rich class of expressible, natural, and useful submodular cost models. We empirically demonstrate the utility of our models on a real world problem of cooperative image matching.