Streaming Submodular Maximization under Matroid Constraints

TL;DR

This paper develops new streaming algorithms for maximizing monotone submodular functions under matroid constraints, achieving improved approximation guarantees with limited memory and multiple passes, advancing theoretical understanding in this area.

Contribution

Introduces single-pass and multi-pass streaming algorithms for matroid constraints with near-optimal approximation guarantees, filling a key gap in streaming submodular maximization research.

Findings

Single-pass algorithm with 0.3178 approximation

Multi-pass algorithm with (1-1/e - ε) approximation

Tight bounds showing limitations of better guarantees

Abstract

Recent progress in (semi-)streaming algorithms for monotone submodular function maximization has led to tight results for a simple cardinality constraint. However, current techniques fail to give a similar understanding for natural generalizations, including matroid constraints. This paper aims at closing this gap. For a single matroid of rank $k$ (i.e., any solution has cardinality at most $k$), our main results are: 1) a single-pass streaming algorithm that uses $\widetilde{O}(k)$ memory and achieves an approximation guarantee of $0.3178$, and 2) a multi-pass streaming algorithm that uses $\widetilde{O}(k)$ memory and achieves an approximation guarantee of $(1-1/e - \varepsilon)$ by taking a constant (depending on $\varepsilon$) number of passes over the stream. This improves on the previously best approximation guarantees of $1/4$ and $1/2$ for single-pass and multi-pass streaming algorithms, respectively. In fact, our multi-pass streaming algorithm is tight in that any algorithm with a better guarantee than $1/2$ must make several passes through the stream and any algorithm that beats our guarantee of $1-1/e$ must make linearly many passes (as well as an exponential number of value oracle queries). Moreover, we show how the approach we use for multi-pass streaming can be further strengthened if the elements of the stream arrive in uniformly random order, implying an improved result for $p$-matchoid constraints.