Parameterized Streaming Algorithms for Vertex Cover

TL;DR

This paper introduces the first streaming algorithms for the parameterized Vertex Cover problem across various dynamic graph models, combining kernelization and randomized sketches, with proven space complexity bounds.

Contribution

It presents novel parameterized streaming algorithms for Vertex Cover in insertion-only, dynamic, and promised dynamic models, using kernelization and randomized sketches.

Findings

First algorithms for parameterized Vertex Cover in streaming models

Matching lower bounds on space complexity

Effective handling of dynamic graph streams with solutions bounded by parameter k

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 where 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 the Vertex Cover problem. We consider the following three models for a graph stream on $n$ nodes: 1. The insertion-only model where the edges can only be added. 2. The dynamic model where edges can be both inserted and deleted. 3. The \emph{promised} dynamic model where we are guaranteed that at each timestamp there is a solution of size at most $k$. In each of these three models we are able to design parameterized streaming algorithms for the Vertex Cover problem. We are also able to show matching lower bound for the space complexity of our algorithms. (Due to the arXiv limit of 1920 characters for abstract field, please see the abstract in the paper for detailed description of our results)