Streaming Maximal Matching with Bounded Deletions

We initiate the study of the Maximal Matching problem in bounded-deletion graph streams. In this setting, a graph $G$ is revealed as an arbitrary sequence of edge insertions and deletions, where the number of insertions is unrestricted but the number of deletions is guaranteed to be at most $K$, for some given parameter $K$. The single-pass streaming space complexity of this problem is known to be $\Theta(n^2)$ when $K$ is unrestricted, where $n$ is the number of vertices of the input graph. In this work, we present new randomized and deterministic algorithms and matching lower bound results that together give a tight understanding (up to poly-log factors) of how the space complexity of Maximal Matching evolves as a function of the parameter $K$: The randomized space complexity of this problem is $\tilde{\Theta}(n \cdot \sqrt{K})$, while the deterministic space complexity is $\tilde{\Theta}(n \cdot K)$. We further show that if we relax the maximal matching requirement to an $\alpha$-approximation to Maximum Matching, for any constant $\alpha > 2$, then the space complexity for both, deterministic and randomized algorithms, strikingly changes to $\tilde{\Theta}(n + K)$.