Cardinality constrained submodular maximization for random streams

TL;DR

This paper improves algorithms for maximizing submodular functions under cardinality constraints in streaming models, achieving better memory efficiency and extending to non-monotone functions, with practical implementation success.

Contribution

It simplifies and enhances existing algorithms for submodular maximization in random streams, reducing memory use and extending applicability to non-monotone functions.

Findings

Achieved $O(k/\varepsilon)$ memory complexity for monotone functions.

Provided a $(1/e-\varepsilon)$-approximation for non-monotone functions.

Validated algorithms on real-world datasets.

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 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.