# Planar Graph Perfect Matching Is in NC

@article{Anari2017PlanarGP,
title={Planar Graph Perfect Matching Is in NC},
author={Nima Anari and Vijay V. Vazirani},
journal={2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)},
year={2017},
pages={650-661}
}
• Published 22 September 2017
• Computer Science
• 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
Is perfect matching in NC? That is, is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in theoretical computer science for over three decades, ever since the discovery of RNC matching algorithms. Within this question, the case of planar graphs has remained an enigma: On the one hand, counting the number of perfect matchings is far harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs…

## Figures from this paper

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This paper obtains NC algorithms for perfect matching in any minor-closed graph family that forbids a one-crossing graph and defines and uses matching-mimicking networks, small replacement networks that behave the same, with respect to matching problems involving a fixed set of terminals, as the larger network they replace.
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This article gives an almost complete derandomization of the Isolation Lemma for perfect matchings in bipartite graphs and presents three different ways of doing this construction with a common main idea.
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The gap between bipartite and general graphs is bridged, by giving an O ε (poly(log n )) update time algorithm that maintains a (1 + ε )-approximate maximum integral matching under adversarial deletions under partially dynamic matching.

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This work alters the algorithm of Gallucio and Loebl to show that counting the number of perfect matchings in graphs of small genus is in NC, and rekindles the hope for an NC-algorithm to find a perfect matching in a non-bipart i te planar graph.
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We show that for any class of bipartite graphs which is closed under edge deletion and where the number of perfect matchings can be counted in NC, there is a deterministic NC algorithm for finding a
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This work gives what appears to be the culmination of this line of work: An NC algorithm for finding a minimum-weight perfect matching in a general graph with polynomially bounded edge weights, provided it is given an oracle for the decision problem.
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2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
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It is shown that the perfect matching problem in general graphs is in Quasi-NC, and the result is obtained by a derandomization of the Isolation Lemma for perfect matchings to obtain a Randomized NC algorithm.
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This paper obtains NC algorithms for perfect matching in any minor-closed graph family that forbids a one-crossing graph and defines matching-mimicking networks, small replacement networks that behave the same, with respect to matching problems involving a fixed set of terminals, as the larger network they replace.
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We show that the problem of constructing a perfect matching in a graph is in the complexity class Random NC; i.e., the problem is solvable in polylog time by a randomized parallel algorithm using a
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The main new idea enabling the results is the definition and use of matching-mimicking networks, small replacement networks that behave the same, with respect to matching problems involving a fixed set of terminals, as the larger network they replace.
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The algorithm is the first algorithm to return unique perfect matchings with only polynomially many processors, and is also the first pseudo-deterministic RNC algorithm for depth first search (DFS).
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