Multi-Pass Streaming Algorithms for Monotone Submodular Function Maximization

TL;DR

This paper introduces multi-pass streaming algorithms for maximizing monotone submodular functions under cardinality and knapsack constraints, achieving near-optimal approximation ratios efficiently in terms of time and space.

Contribution

The paper presents novel multi-pass streaming algorithms with approximation guarantees for submodular maximization under constraints, improving runtime over previous methods.

Findings

Achieves a (1 - e^{-1} - ε)-approximation for cardinality constraints.

Achieves a (0.5 - ε)-approximation for knapsack constraints.

Runs in near-linear time with respect to input size, using limited memory.

Abstract

We consider maximizing a monotone submodular function under a cardinality constraint or a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access to only a small fraction of the data stored in primary memory. We propose the following streaming algorithms taking $O(\varepsilon^{-1})$ passes: ----a $(1-e^{-1}-\varepsilon)$-approximation algorithm for the cardinality-constrained problem ---- a $(0.5-\varepsilon)$-approximation algorithm for the knapsack-constrained problem. Both of our algorithms run in $O^\ast(n)$ time, using $O^\ast(K)$ space, where $n$ is the size of the ground set and $K$ is the size of the knapsack. Here the term $O^\ast$ hides a polynomial of $\log K$ and $\varepsilon^{-1}$. Our streaming algorithms can also be used as fast approximation algorithms. In particular, for the cardinality-constrained problem, our algorithm takes $O(n\varepsilon^{-1} \log (\varepsilon^{-1}\log K) )$ time, improving on the algorithm of Badanidiyuru and Vondr\'{a}k that takes $O(n \varepsilon^{-1} \log (\varepsilon^{-1} K) )$ time.