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Revision #1 to TR17-059 | 5th September 2017 15:50

The Matching Problem in General Graphs is in Quasi-NC

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Revision #1
Authors: Ola Svensson, Jakub Tarnawski
Accepted on: 5th September 2017 15:51
Downloads: 9
Keywords: 


Abstract:

We show that the perfect matching problem in general graphs is in Quasi-NC. That is, we give a deterministic parallel algorithm which runs in $O(\log^3 n)$ time on $n^{O(\log^2 n)}$ processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to obtain a Randomized NC algorithm.

Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.


Paper:

TR17-059 | 6th April 2017 20:03

The Matching Problem in General Graphs is in Quasi-NC





TR17-059
Authors: Ola Svensson, Jakub Tarnawski
Publication: 9th April 2017 10:10
Downloads: 258
Keywords: 


Abstract:

We show that the perfect matching problem in general graphs is in Quasi-NC. That is, we give a deterministic parallel algorithm which runs in $O(\log^3 n)$ time on $n^{O(\log^2 n)}$ processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to obtain a Randomized NC algorithm.

Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.



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