Constrained Robust Submodular Optimization

In this paper, we consider the problem of constrained maximization of the minimum of a set of submodular functions, in which the goal is to find solutions that are robust to worst-case values of the objective functions. Unfortunately, this problem is both non-submodular and inapproximable. In the case where the submodular functions are monotone, an approximate solution can be found by relaxing the problem. We propose an algorithm called GENERALIZED SATURATE (GENSAT) that exploits the submodular structure of the problem and, as a result, returns a nearoptimal solution with a constant-factor approximation guarantee on the relaxed problem. GENSAT can handle any submodular constraint, e.g. matroids, cover, and knapsack, and is compatible with any submodular maximization algorithm.