On the complexity of dynamic submodular maximization

DOI10.1145/3519935.3519951arXiv2111.03198OpenAlexW3214642224MaRDI QIDQ6083622

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Publication date: 8 December 2023

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Abstract: 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 $(0.5+epsilon)$-approximate solution under a cardinality constraint, for any constant $epsilon>0$, 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 $(0.5-epsilon)$-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 $1-1/e-epsilon$ and amortized query complexities $smashO(log(k/epsilon)/epsilo{n}^2)$ and $smash{k}^{ildeO(1/epsilo{n}^2)}logn$, respectively, where $k$ denotes the cardinality parameter or the rank of the matroid.

Full work available at URL: https://arxiv.org/abs/2111.03198

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