Editing Open Problems:60
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{{Header | {{Header | ||
| + | |title=Single-Pass Unweighted Matchings | ||
|source=dortmund12 | |source=dortmund12 | ||
|who=Andrew McGregor | |who=Andrew McGregor | ||
}} | }} | ||
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 {{cite|Kapralov-12}} showed that achieving better than a $1-1/e$ approximation is impossible. If the stream is randomly ordered, Konrad et al. {{cite|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 {{cite|AhnG-11}}) or the best approximation possible for maximum weighted matching in a single pass (see, e.g., Epstein et al. {{cite|EpsteinLMS-11}}). | 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 {{cite|Kapralov-12}} showed that achieving better than a $1-1/e$ approximation is impossible. If the stream is randomly ordered, Konrad et al. {{cite|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 {{cite|AhnG-11}}) or the best approximation possible for maximum weighted matching in a single pass (see, e.g., Epstein et al. {{cite|EpsteinLMS-11}}). | ||
