Space Optimal Vertex Cover in Dynamic Streams

TL;DR

This paper presents a space-optimal randomized algorithm for approximating the minimum vertex cover in dynamic graph streams, matching the known lower bounds and advancing understanding in dynamic graph streaming complexity.

Contribution

The authors develop a randomized algorithm that achieves optimal space complexity for $eta$-approximate minimum vertex cover in dynamic streams, matching the lower bound up to constants.

Findings

Achieves $O(n^2/eta^2)$ bits of space, matching the lower bound.

Identifies 'easy' instances solvable by full vertex set, and 'hard' sparse instances for efficient approximation.

Advances the theoretical understanding of space complexity for dynamic graph problems.

Abstract

We optimally resolve the space complexity for the problem of finding an $\alpha$-approximate minimum vertex cover ($\alpha$MVC) in dynamic graph streams. We give a randomised algorithm for $\alpha$MVC which uses $O(n^2/\alpha^2)$ bits of space matching Dark and Konrad's lower bound [CCC 2020] up to constant factors. By computing a random greedy matching, we identify `easy' instances of the problem which can trivially be solved by returning the entire vertex set. The remaining `hard' instances, then have sparse induced subgraphs which we exploit to get our space savings and solve $\alpha$MVC. Achieving this type of optimality result is crucial for providing a complete understanding of a problem, and it has been gaining interest within the dynamic graph streaming community. For connectivity, Nelson and Yu [SODA 2019] improved the lower bound showing that $\Omega(n \log^3 n)$ bits of space is necessary while Ahn, Guha, and McGregor [SODA 2012] have shown that $O(n \log^3 n)$ bits is sufficient. For finding an $\alpha$-approximate maximum matching, the upper bound was improved by Assadi and Shah [ITCS 2022] showing that $O(n^2/\alpha^3)$ bits is sufficient while Dark and Konrad [CCC 2020] have shown that $\Omega(n^2/\alpha^3)$ bits is necessary. The space complexity, however, remains unresolved for many other dynamic graph streaming problems where further improvements can still be made. \end{abstract}