Submodular Maximization Meets Streaming: Matchings, Matroids, and More

TL;DR

This paper introduces two semi-streaming algorithms for maximum submodular-function matching, extending classical matching algorithms to a broader submodular setting with provable approximation guarantees.

Contribution

It presents the first semi-streaming algorithms with approximation guarantees for MSM, generalizing maximum weight matching algorithms to submodular functions and complex independence systems.

Findings

Achieves a 7.75-approximation in one pass

Achieves a (3+ε)-approximation in multiple passes

Extends results to hypermatchings and matroid intersections

Abstract

We study the problem of finding a maximum matching in a graph given by an input stream listing its edges in some arbitrary order, where the quantity to be maximized is given by a monotone submodular function on subsets of edges. This problem, which we call maximum submodular-function matching (MSM), is a natural generalization of maximum weight matching (MWM), which is in turn a generalization of maximum cardinality matching (MCM). We give two incomparable algorithms for this problem with space usage falling in the semi-streaming range---they store only $O(n)$ edges, using $O(n\log n)$ working memory---that achieve approximation ratios of $7.75$ in a single pass and $(3+\epsilon)$ in $O(\epsilon^{-3})$ passes respectively. The operations of these algorithms mimic those of Zelke's and McGregor's respective algorithms for MWM; the novelty lies in the analysis for the MSM setting. In fact we identify a general framework for MWM algorithms that allows this kind of adaptation to the broader setting of MSM. In the sequel, we give generalizations of these results where the maximization is over "independent sets" in a very general sense. This generalization captures hypermatchings in hypergraphs as well as independence in the intersection of multiple matroids.