On the Complexity of Dynamic Submodular Maximization

TL;DR

This paper investigates the computational complexity of maintaining approximate solutions for dynamic submodular maximization under insertions and deletions, revealing fundamental limitations and proposing efficient algorithms for insertion-only streams.

Contribution

It establishes lower bounds on query complexity for dynamic algorithms with certain approximation ratios and introduces efficient algorithms for insertion-only streams with near-optimal guarantees.

Findings

Any dynamic algorithm with >0.5 approximation requires polynomial query complexity in n.

Achieving 0.584-approximation needs linear query complexity.

Efficient algorithms with near-optimal guarantees for insertion-only streams.

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.