Title:

Streaming Algorithms for Estimating the Matching Size in Planar Graphs and Beyond.

Abstract:

We consider the problem of estimating the size of a maximum matching when the edges are revealed in a streaming fashion. Consider a graph G=(V,E) with n vertices and m edges. The input stream is a permutation of edges S= (e_1,…,e_m) chosen by an adversary.
The goal is to output an estimation of the size of a maximum matching. The algorithm is only allowed to use a small amount of memory (much smaller than n).

When the underlying graph is planar, we present a simple and elegant streaming algorithm that with high probability estimates the size of a maximum matching within a constant factor using O-tilde(n^(2/3)) space. The approach generalizes to the family of graphs that have bounded arboricity. Graphs with bounded arboricity include, among other families of graphs, graphs with an excluded constant-size minor. To the best of our knowledge, this is the first result for estimating the size of a maximum matching in the adversarial-order streaming model (as opposed to the random-order streaming model). We circumvent the barriers inherent in the adversarial-order model by exploiting several structural properties of planar graphs, and more generally, graphs with bounded arboricity. We hope that this approach finds applications in estimating other properties of graphs in the adversarial-order streaming model. We further reduce the required memory size to O-tilde(sqrt(n)) for three restricted settings: (i) when the underlying graph is a forest; (ii) when we have 2-passes over the stream of edges of a graph with bounded arboricity; and (iii) when the edges arrive in random order and the underlying graph has bounded arboricity.

Finally, by introducing a communication complexity problem, we show that the approximation factor of a deterministic algorithm cannot be better than a constant using o(n) space, even if the underlying graph is a collection of paths. We can show that under a plausible conjecture for the hardness of the communication complexity problem, randomized algorithms with o(sqrt(n)) space cannot have an approximation factor better than a fixed constant.