Parameterized Streaming: Maximal Matching and Vertex Cover

In the insertion only model, there is a one-pass deterministic algorithm for the parameterized Vertex Cover problem which computes a sketch using Õ(k²) space¹ such that at each timestamp in time Õ(2^k) it can either extract a solution of size at most k for the current instance, or report that no such solution exists. We also show a tight lower bound of Ω(k²) for the space complexity of any (randomized) streaming algorithms for the parameterized Vertex Cover, even in the insertion-only model.

In the dynamic model, and under the promise that at each timestamp there is a maximal matching of size at most k, there is a one-pass Õ(k²)-space (sketch-based) dynamic algorithm that maintains a maximal matching with worst-case update time Õ(k²). This algorithm partially solves Open Problem 64 from [1]. An application of this dynamic matching algorithm is a one-pass Õ(k²)-space streaming algorithm for the parameterized Vertex Cover problem that in time Õ(2^k) extracts a solution for the final instance with probability 1 – δ/n^o(1), where δ < 1. To the best of our knowledge, this is the first graph streaming algorithm that combines linear sketching with sequential operations that depend on the graph at the current time.

In the dynamic model without any promise, there is a one-pass randomized algorithm for the parameterized Vertex Cover problem which computes a sketch using Õ(nk) space such that in time Õ(nk + 2^k) it can either extract a solution of size at most k for the final instance, or report that no such solution exists.