On the Complexity of Dynamic Submodular Maximization
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We study dynamic algorithms for the problem of maximizing a monotone submodular function over a stream of n insertions and deletions. We show that any algorithm that maintains a (0.5+ϵ)-approximate solution under a cardinality constraint, for any constant ϵ>0, must have an amortized query complexity that is 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 in n. Moreover, a linear amortized query complexity is needed in order to maintain a 0.584-approximate solution. This is in sharp contrast with recent dynamic algorithms of [LMNF+20, Mon20] that achieve (0.5-ϵ)-approximation with a 𝗉𝗈𝗅𝗒log(n) amortized query complexity. On the positive side, when the stream is insertion-only, we present efficient algorithms for the problem under a cardinality constraint and under a matroid constraint with approximation guarantee 1-1/e-ϵ and amortized query complexities O(log (k/ϵ)/ϵ^2) and k^Õ(1/ϵ^2)log n, respectively, where k denotes the cardinality parameter or the rank of the matroid.
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