On the Complexity of Dynamic Submodular Maximization

• URL: http://arxiv.org/abs/2111.03198v1
• Date: Fri, 5 Nov 2021 00:04:29 GMT
• Title: On the Complexity of Dynamic Submodular Maximization
• Authors: Xi Chen, Binghui Peng
• Abstract summary: 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 $n$. 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.
• Score: 15.406670088500087
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\mathit{polynomial}$ 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 $(0.5-\epsilon)$-approximation with a $\mathsf{poly}\log(n)$ 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 $\smash{O(\log (k/\epsilon)/\epsilon^2)}$ and $\smash{k^{\tilde{O}(1/\epsilon^2)}\log n}$, respectively, where $k$ denotes the cardinality parameter or the rank of the matroid.

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