Quick Streaming Algorithms for Maximization of Monotone Submodular Functions in Linear Time

TL;DR

This paper introduces the first deterministic linear-time streaming algorithms for monotone submodular maximization under a cardinality constraint, achieving near-optimal ratios with fewer queries and demonstrating superior empirical performance.

Contribution

The paper presents the first deterministic linear-time streaming algorithms for monotone submodular maximization, including single-pass and multi-pass variants with provable guarantees.

Findings

Algorithms require fewer queries than existing methods

Achieve better objective values empirically

Establish lower bounds on query complexity

Abstract

We consider the problem of monotone, submodular maximization over a ground set of size $n$ subject to cardinality constraint $k$. For this problem, we introduce the first deterministic algorithms with linear time complexity; these algorithms are streaming algorithms. Our single-pass algorithm obtains a constant ratio in $\lceil n / c \rceil + c$ oracle queries, for any $c \ge 1$. In addition, we propose a deterministic, multi-pass streaming algorithm with a constant number of passes that achieves nearly the optimal ratio with linear query and time complexities. We prove a lower bound that implies no constant-factor approximation exists using $o(n)$ queries, even if queries to infeasible sets are allowed. An empirical analysis demonstrates that our algorithms require fewer queries (often substantially less than $n$) yet still achieve better objective value than the current state-of-the-art algorithms, including single-pass, multi-pass, and non-streaming algorithms.