Space Optimal Vertex Cover in Dynamic Streams

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}