Problem 60: Single-Pass Unweighted Matchings

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Suggested by Andrew McGregor
Source Dortmund 2012
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Suppose you have $O(n \operatorname{polylog} n)$ memory and a single pass over a stream of $m$ edges (arbitrarily ordered) on $n$ nodes. How well can you approximate the size of the maximum cardinality matching? A trivial greedy algorithm finds a $1/2$-approximation but that's still the best known algorithm in the general setting. Kapralov [Kapralov-12] showed that achieving better than a $1-1/e$ approximation is impossible. If the stream is randomly ordered, Konrad et al. [KonradMM-12] presented a $1/2 + 0.005$-approximation. Other variants of the question are also open, e.g., achieving a $(1-\epsilon)$ approximation in the minimum number of passes (see, e.g., Ahn and Guha [AhnG-11]) or the best approximation possible for maximum weighted matching in a single pass (see, e.g., Epstein et al. [EpsteinLMS-11]).