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In this paper, we investigate a class of submodular problems which in generalare very hard. These include minimizing a submodular cost function undercombinatorial constraints, which include cuts, matchings, paths, etc.,optimizing a submodular function under submodular cover and submodular knapsackconstraints, and minimizing a ratio of submodular functions. All these problemsappear in several real world problems but have hardness factors of$\Omega(\sqrt{n})$ for general submodular cost functions. We show how we canachieve constant approximation factors when we restrict the cost functions tolow rank sums of concave over modular functions. A wide variety of machinelearning applications are very naturally modeled via this subclass ofsubmodular functions. Our work therefore provides a tighter connection betweentheory and practice by enabling theoretically satisfying guarantees for a richclass of expressible, natural, and useful submodular cost models. Weempirically demonstrate the utility of our models on real world problems ofcooperative image matching and sensor placement with cooperative costs.

Near Optimal Algorithms for Hard Submodular Programs with Discounted Cooperative Costs