# 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} }

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…

## 22 Citations

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

- Computer ScienceITCS
- 2020

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.

### A Pseudo-Deterministic RNC Algorithm for General Graph Perfect Matching

- Computer ScienceArXiv
- 2019

An NC reduction from search to decision for the problem of finding a minimum weight perfect matching, provided edge weights are polynomially bounded, and a corollary of this reduction is an analogous algorithm for general graphs.

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

- MathematicsSPAA
- 2019

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.

### Finding big matchings in planar graphs quickly

- Mathematics, Computer ScienceArXiv
- 2019

This paper gives a linear-time algorithm that finds a matching of size at least $\frac{n}{3}$ in any planar graph with minimum degree 3.

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

- MathematicsSIAM J. Comput.
- 2021

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.

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

- MathematicsArXiv
- 2018

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.

### Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm

- Mathematics, Computer ScienceISAAC
- 2018

It is shown that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs.

### Planarity, Exclusivity, and Unambiguity

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2019

New upper bounds on the complexity of the s-t-connectivity problem in planar graphs are provided, thereby providing additional evidence that this problem is not complete for NL and a new upper bound on thecomplexity of computing edit distance is provided.

### A deterministic parallel algorithm for bipartite perfect matching

- Mathematics, Computer ScienceCommun. ACM
- 2019

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.

### Decremental Matching in General Graphs

- Mathematics, Computer ScienceICALP
- 2022

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 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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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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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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