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}
}
  • Nima AnariV. Vazirani
  • 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… 

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References

SHOWING 1-10 OF 48 REFERENCES

A new NC-algorithm for finding a perfect matching in bipartite planar and small genus graphs (extended abstract)

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.

Some perfect matchings and perfect half-integral matchings in NC

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

Matching Is as Easy as the Decision Problem, in the NC Model

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.

The Matching Problem in General Graphs Is in Quasi-NC

  • O. SvenssonJakub Tarnawski
  • Mathematics, Computer Science
    2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2017
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.

NC Algorithms for Computing a Perfect Matching, the Number of Perfect Matchings, and a Maximum Flow in One-Crossing-Minor-Free Graphs

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.

Constructing a perfect matching is in random NC

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

NC Algorithms for Perfect Matching and Maximum Flow in One-Crossing-Minor-Free Graphs

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.

Perfect Bipartite Matching in Pseudo-Deterministic RNC

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).

Maximum matching and a polyhedron with 0,1-vertices

The emphasis in this paper is on relating the matching problem to the theory of continuous linear programming, and the algorithm described does not involve any "blind-alley programming" -which, essentially, amounts to testing a great many combinations.