# Problem 64: Matchings in the Turnstile Model

Suggested by | Andrew McGregor |
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Source | Bertinoro 2014 |

Short link | https://sublinear.info/64 |

Consider an unweighted graph on $n$ nodes defined by a stream of edge insertions and deletions. Is it possible to approximate the size of the maximum cardinality matching up to constant factor given a single pass and $o(n^2)$ space? Recall that a factor 2 approximation is easy in $O(n \log n)$ space if there are no edge deletions.

## Update

The question is fully settled when the goal is to output the edges of an approximate maximum matching: to obtain an $\alpha$-approximation to maximum matching in dynamic streams, $\Omega(n^2/\alpha^3)$ space is necessary [AssadiKLY-16] and $\widetilde{O}(n^2/\alpha^3)$ space is sufficient [AssadiKLY16,ChitnisCEHMMV-16]. When the goal is only to estimate the value of maximum matching (as opposed to finding the edges), it was shown in [AssadiKL-17] that $\Omega(n/\alpha^2)$ space is necessary and $\widetilde{O}(n^2/\alpha^4)$ space is sufficient.