Robust monotone submodular function maximization

We consider a robust formulation, introduced by Krause et al. (J Artif Intell Res 42:427–486, 2011), of the classical cardinality constrained monotone submodular function maximization problem, and give the first constant factor approximation results. The robustness considered is w.r.t. adversarial removal of up to τ elements from the chosen set. For the fundamental case of τ= 1 , we give a deterministic (1 - 1 / e) - 1 / Θ(m) approximation algorithm, where m is an input parameter and number of queries scale as O(n^{m + 1}). In the process, we develop a deterministic (1 - 1 / e) - 1 / Θ(m) approximate greedy algorithm for bi-objective maximization of (two) monotone submodular functions. Generalizing the ideas and using a result from Chekuri et al. (in: FOCS 10, IEEE, pp 575–584, 2010), we show a randomized (1 - 1 / e) - ϵ approximation for constant τ and ϵ≤1Ω~(τ), making O(n1/ϵ3) queries. Further, for τ≪k, we give a fast and practical 0.387 algorithm. Finally, we also give a black box result result for the much more general setting of robust maximization subject to an Independence System.