# The Matching Problem in General Graphs Is in Quasi-NC

@article{Svensson2017TheMP, title={The Matching Problem in General Graphs Is in Quasi-NC}, author={Ola Svensson and Jakub Tarnawski}, journal={2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)}, year={2017}, pages={696-707} }

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…

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## 47 Citations

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## References

SHOWING 1-10 OF 57 REFERENCES

The Polynomially Bounded Perfect Matching Problem Is in NC 2

- MathematicsSTACS
- 2006

It is shown that for any graph that has a polynomially bounded number of perfect matchings, it is possible to construct allperfect matchings in NC2, and this result is extended to weighted graphs.

The matching problem for bipartite graphs with polynomially bounded permanents is in NC

- Mathematics, Computer Science28th Annual Symposium on Foundations of Computer Science (sfcs 1987)
- 1987

An NC3 algorithm for the problem of constructing all perfect matchings in a graph G with a permanent bounded by O(nk) is designed, which entails also among other things an efficient NC3-algorithm for computing small (polynomially bounded) arithmetic permanents, and a sublinear parallel time algorithm for enumerating all the perfect matching in graphs with permanents up to 2nε.

On the Parallel Complexity of Hamiltonian Cycle and Matching Problem on Dense Graphs

- Mathematics, Computer ScienceJ. Algorithms
- 1993

It is proved that finding an NC algorithm for perfect matching in slightly less dense graphs (minimum degree is at least (12 ? ?)|V|) is as hard as the same problem for all graphs, and interestingly the problem of finding a Hamiltonian cycle becomes NP-complete.

Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs

- Computer Science, MathematicsTheory of Computing Systems
- 2009

A deterministic Logspace procedure, which, given a bipartite planar graph on n vertices, assigns O(log n) bits long weights to its edges so that the minimum weight perfect matching in the graph becomes unique, and tries to find the lower bound on the number of bits needed for deterministically isolating a perfect matching.

Constructing a perfect matching is in random NC

- Mathematics, Computer ScienceComb.
- 1986

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…

Maximum matchings via Gaussian elimination

- Computer Science45th Annual IEEE Symposium on Foundations of Computer Science
- 2004

The results resolve a long-standing open question of whether Lovasz's randomized technique of testing graphs for perfect matching in time O(n/sup w/) can be extended to an algorithm that actually constructs a perfect matching.

Bipartite perfect matching is in quasi-NC

- MathematicsSTOC
- 2015

We show that the bipartite perfect matching problem is in quasi- NC2. That is, it has uniform circuits of quasi-polynomial size nO(logn), and O(log2 n) depth. Previously, only an exponential upper…

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

- Computer ScienceSTOC '00
- 2000

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.

NC Algorithms for Computing the Number of Perfect Matchings in K3, 3-free Graphs and Related Problems

- Computer Science, MathematicsSWAT
- 1988

It is shown that the problem of computing the number of perfect matchings in K3,3-free graphs is in NC, and this result opens up the possibility of obtaining an NC algorithm for finding a perfect matching in K2,2- free graphs.

Linear matroid intersection is in quasi-NC

- MathematicsElectron. Colloquium Comput. Complex.
- 2016

It is shown that the linear matroid intersection problem is in quasi-NC2, that is, it has uniform circuits of quasi-polynomial size nO(logn), and O(log2 n) depth, which generalizes the similar result for the bipartite perfect matching problem.