Adaptive Robust Submodular Optimization and Beyond

Constrained submodular maximization has been extensively studied in the recent years. In this paper, we study adaptive robust optimization with nearly submodular structure (ARONSS). Our objective is to randomly select a subset of items that maximizes the worst case value of several reward functions simultaneously. Our work differs from existing studies in two ways: (1) we study the robust optimization problem under the adaptive setting, i.e., one needs to adaptively select items based on the feedback collected from picked items, and (2) our results apply to a broad range of reward functions characterized by (Formula Presented)-nearly submodular function. We first analyze the adaptivity gap of ARONSS and show that the gap between the best adaptive solution and the best non-adaptive solution is bounded. Then we propose an approximate solution to this problem when all reward functions are submodular. In particular, our algorithm achieves approximation ratio (Formula Presented) when considering a single matroid constraint. At last, we present two heuristics for the general case with nearly submodular functions. All proposed solutions are non-adaptive which are easy to implement.