On the Complexity of Dynamic Submodular Maximization

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 mathitpolynomial 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 mathsfpolylog(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 smashO(log (k/ε)/ε2) and smashk^tildeO(1/ε2)log n, respectively, where k denotes the cardinality parameter or the rank of the matroid.