Parameterized streaming : maximal matching and vertex cover

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Abstract

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 O~(k2) space [ O~(f(k))=O(f(k)·logO(1) m), where m is the number of edges.] such that at each timestamp in time ~O(2k) 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 O~(k2)-space (sketch-based) dynamic algorithm that maintains a maximal matching with worst-case update time [The time to update the current maximal matching upon an insertion or deletion.] O~(k2). This algorithm partially solves Open Problem 64 from the List of open problems in sublinear algorithms. An application of this dynamic matching algorithm is a one-pass O~(k2)-space streaming algorithm for the parameterized Vertex Cover problem that in time O~(2k) extracts a solution for the final instance with probability 1-δ/nO(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 O~(nk) space such that in time O~(nk+2k) it can either extract a solution of size at most k for the final instance, or report that no such solution exists.

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