Maximum Matchings in Dynamic Graph Streams and the Simultaneous Communication Model.

We study the problem of finding an approximate maximum matching in two closely related computational models, namely, the dynamic graph streaming model and the simultaneous multi-party communication model. In the dynamic graph streaming model, the input graph is revealed as a stream of edge insertions and deletions, and the goal is to design a small space algorithm to approximate the maximum matching. In the simultaneous model, the input graph is partitioned across k players, and the goal is to design a protocol where the k players simultaneously send a small-size message to a coordinator, and the coordinator computes an approximate matching. Dynamic graph streams. We resolve the space complexity of single-pass turnstile streaming algorithms for approximating matchings by showing that for any ε > 0, Θ(n2-3ε) space is both sufficient and necessary (up to polylogarithmic factors) to compute an nε-approximate matching; here n denotes the number of vertices in the input graph. The simultaneous communication model. Our results for dynamic graph streams also resolve the (per-player) simultaneous communication complexity for approximating matchings in the edge partition model. For the vertex partition model, we design new randomized and deterministic protocols for k players to achieve an α-approximation. Specifically, for α ≥ [EQUATION], we provide a randomized protocol with total communication of O(nk/α2) and a deterministic protocol with total communication of O(nk/α). Both these bounds are tight. Our work generalizes the results established by Dobzinski et al. (STOC 2014) for the special case of k = n. Finally, for the case of α = o([EQUATION]), we establish a new lower bound on the simultaneous communication complexity which is super-linear in n.