Optimal deletion-robust coreset for submodular maximization
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In recent years we have witnessed an increase on the development of methods for submodular optimization, which have been motivated by the wide applicability of submodular functions in real-world data-science problems. In this paper, we contribute to this line of work by considering the problem of robust submodular maximization against unexpected deletions, which may occur due to privacy issues or user preferences. Specifically, we study and derive bounds for the minimum number of items a streaming algorithm has to remember, in order to achieve a non-trivial approximation guarantee against adversarial deletions up to d items. We call the set of items that are kept by the algorithm before adversarial deletions a deletion-robust coreset. We propose a single-pass streaming algorithm that yields a (1-2ϵ)/(4p)-approximation for maximizing a non-decreasing submodular function under a general p-matroid constraint and requires a coreset size k + d/ϵ, where k is the maximum size of a feasible solution. To the best of our knowledge, this is the first work to achieve an (asymptotically) optimal coreset size, as no constant approximation is possible with a coreset of size sublinear in d.
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