Streaming Submodular Matching Meets the Primal-Dual Method

TL;DR

This paper advances the theoretical understanding of streaming submodular maximization under matching constraints by providing improved approximation algorithms and bounds using primal-dual methods, unifying and extending previous techniques.

Contribution

It introduces primal-dual algorithms for MSM with better approximation ratios and establishes new lower bounds, pioneering the use of primal-dual analysis in streaming submodular optimization.

Findings

Improved approximation ratios for monotone and non-monotone MSM.

New lower bounds for streaming MSM algorithms.

First application of primal-dual methods in streaming submodular maximization.

Abstract

We study streaming submodular maximization subject to matching/$b$-matching constraints (MSM/MSbM), and present improved upper and lower bounds for these problems. On the upper bounds front, we give primal-dual algorithms achieving the following approximation ratios. $\bullet$ $3+2\sqrt{2}\approx 5.828$ for monotone MSM, improving the previous best ratio of $7.75$. $\bullet$ $4+3\sqrt{2}\approx 7.464$ for non-monotone MSM, improving the previous best ratio of $9.899$. $\bullet$ $3+\epsilon$ for maximum weight b-matching, improving the previous best ratio of $4+\epsilon$. On the lower bounds front, we improve on the previous best lower bound of $\frac{e}{e-1}\approx 1.582$ for MSM, and show ETH-based lower bounds of $\approx 1.914$ for polytime monotone MSM streaming algorithms. Our most substantial contributions are our algorithmic techniques. We show that the (randomized) primal-dual method, which originated in the study of maximum weight matching (MWM), is also useful in the context of MSM. To our knowledge, this is the first use of primal-dual based analysis for streaming submodular optimization. We also show how to reinterpret previous algorithms for MSM in our framework; hence, we hope our work is a step towards unifying old and new techniques for streaming submodular maximization, and that it paves the way for further new results.