Maximum Matchings in Dynamic Graph Streams and the Simultaneous Communication Model

Abstract

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 > 0, Θ(n 2−3) space is both sufficient and necessary (up to polylogarithmic factors) to compute an n-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 ≥ √ k, we provide a randomized protocol with total communication of O(nk/α 2) 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(√ k), we establish a new lower bound on the simultaneous communication complexity which is super-linear in n.

Sublinear Algorithms for (Δ + 1) Vertex Coloring

“…No o(n 2 ) space single-pass streaming algorithm was known for this problem even in insertion-only streams. This state-of-affairs was in fact similar to the case of the closely related maximal matching problem: the best known semi-streaming algorithm for this problem on dynamic streams uses Θ(log n) passes [3,35] and it is provably impossible to solve this problem using o(n 2 )-space in a single pass over a dynamic stream [9] (although this problem is trivial in insertion-only streams). Considering this one might have guessed a similar lower bound also holds for the (∆ + 1) coloring problem.…”

Tight Bounds on the Round Complexity of the Distributed Maximum Coverage Problem

“…The proof now consists of two parts: (i) use the simultaneity of the communication to argue that as each machine is oblivious to identity of its special set, it cannot convey enough information about this set using limited communication, and (ii) use the bound on the size of the intersection between the sets to show that this prevents the coordinator to find a good solution. The strategy outlined above is in fact at the core of many existing lower bounds for simultaneous protocols in the coordinator model including [12,13,34,51] (a notable exception is the lower bound of [12] on estimating matching size in sparse graphs). For example, to obtain the hard input distributions in [13,51] for the maximum matching problem, we just need to switch the sets in the small intersecting family above with induced matchings in a Ruzsa-Szemerédi graph [65] (see also [6] for more details on these graphs).…”

A (2 + ∊)-Approximation for Maximum Weight Matching in the Semi-Streaming Model

“…The problem of MWM was also considered in other streaming models, such as the MapReduce model [8,15], the sliding-window model [8,9] and the turnstile stream model (allowing deletions as well as insertions) [1,5,7,14]. More general submodular-function matching problems in the semi-streaming model have been considered by Varadaraja and by Chakrabarti and Kale [6,18].…”