Submodular maximization meets streaming: matchings, matroids, and more

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. 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 $(3+\varepsilon )$ in $O(\varepsilon ^{-3})$ 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. Our framework is not specific to matchings. Rather, we identify a general pattern for algorithms that maximize linear weight functions over “independent sets” and prove that such algorithms can be adapted to maximize a submodular function. The notion of independence here is very general; in particular, appealing to known weight-maximization algorithms, we obtain results for submodular maximization over hypermatchings in hypergraphs as well as independent sets in the intersection of multiple matroids.