Parameterized Streaming: Maximal Matching and Vertex Cover

As graphs continue to grow in size, we seek ways to effectively process such data at scale. The model of streaming graph processing, in which a compact summary is maintained as each edge insertion/deletion is observed, is an attractive one. However, few results are known for optimization problems over such dynamic graph streams. In this paper, we introduce a new approach to handling graph streams, by instead seeking solutions for the parameterized versions of these problems. Here, we are given a parameter k and the objective is to decide whether there is a solution bounded by k. By combining kernelization techniques with randomized sketch structures, we obtain the first streaming algorithms for the parameterized versions of Maximal Matching and Vertex Cover. We consider various models for a graph stream on n nodes: the insertion-only model where the edges can only be added, and the dynamic model where edges can be both inserted and deleted. More formally, we show the following results: • In the insertion only model, there is a one-pass deterministic algorithm for the parameterized Vertex Cover problem which computes a sketch using ∗An earlier draft of this paper was made available online as http://arxiv.org/abs/1405.0093 †Department of Computer Science , University of Maryland at College Park, USA. [email protected]. Supported in part by NSF CAREER award 1053605, NSF grant CCF-1161626, ONR YIP award N000141110662, DARPA/AFOSR grant FA9550-121-0423 and a Simons Award for Graduate Students in Theoretical Computer Science. ‡Department of Computer Science, University of Warwick, UK. [email protected]. Supported in part by the Yahoo Faculty Research and Engagement Program and a Royal Society Wolfson Research Merit Award. §Department of Computer Science , University of Maryland, USA. [email protected]. Supported in part by NSF CAREER award 1053605, NSF grant CCF-1161626, ONR YIP award N000141110662, and DARPA/AFOSR grant FA9550-12-1-0423. ¶Goethe-Universität Frankfurt, Germany and Department of Computer Science, University of Maryland at College Park, USA. [email protected], Supported in part by MO 2200/1-1. Õ(k) space 1 such that at each timestamp in time Õ(2) 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(k) 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(k)space (sketch-based) dynamic algorithm that maintains a maximal matching with worst-case update time Õ(k). This algorithm partially solves Open Problem 64 from [1]. An application of this dynamic matching algorithm is a one-pass Õ(k)space streaming algorithm for the parameterized Vertex Cover problem that in time Õ(2) extracts a solution for the final instance with probability 1−δ/n, 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) it can either extract a solution of size at most k for the final instance, or report that no such solution exists.