Cardinality constrained submodular maximization for random streams

Paul Liu, Aviad Rubinstein, Jan Vondrak, Junyao Zhao

Keywords: submodular, streaming model, optimization, cardinality constraint, approximation algorithms

TL;DR: We give $(1-1/e-\varepsilon)$ and $1/e-\varepsilon$ approximations for submodular maximization under a cardinality constraint in $O(k/\varepsilon)$ memory.

Abstract:

We consider the problem of maximizing submodular functions in single-pass streaming and secretaries-with-shortlists models, both with random arrival order.
For cardinality constrained monotone functions, Agrawal, Shadravan, and Stein~\cite{SMC19} gave a single-pass $(1-1/e-\varepsilon)$-approximation algorithm using only linear memory, but their exponential dependence on $\varepsilon$ makes it impractical even for $\varepsilon=0.1$.
We simplify both the algorithm and the analysis, obtaining an exponential improvement in the $\varepsilon$-dependence (in particular, $O(k/\varepsilon)$ memory).
Extending these techniques, we also give a simple $(1/e-\varepsilon)$-approximation for non-monotone functions in $O(k/\varepsilon)$ memory. For the monotone case, we also give a corresponding unconditional hardness barrier of $1-1/e+\varepsilon$ for single-pass algorithms in randomly ordered streams, even assuming unlimited computation.

Finally, we show that the algorithms are simple to implement and work well on real world datasets.

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Code: https://github.com/where-is-paul/submodular-streaming