# Bipartite perfect matching is in quasi-NC

@article{Fenner2015BipartitePM, title={Bipartite perfect matching is in quasi-NC}, author={Stephen A. Fenner and Rohit Gurjar and Thomas Thierauf}, journal={Proceedings of the forty-eighth annual ACM symposium on Theory of Computing}, year={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 bound was known on the size of such circuits with poly-logarithmic depth. We obtain our result by an almost complete derandomization of the famous Isolation Lemma when applied to yield an efficient randomized parallel algorithm for the bipartite perfect matching problem.

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## 64 Citations

Guest Column: Parallel Algorithms for Perfect Matching

- MathematicsSIGA
- 2017

An almost complete derandomization of the Isolation Lemma for perfect matchings in bipartite graphs is given, giving a deterministic quasi-NC-algorithm for the bipartites perfect matching problem.

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- 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.

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- 2019

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- 2015

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).

Algebraic Representations of Unique Bipartite Perfect Matching

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It is shown that unique bipartite matching is evasive for classical decision trees, and nearly evasive even for generalized query models, and this result extends even to other families of matching-related functions.

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- 2017

It is shown that the answer is that d is linear in n, and the upper bound is an explicit construction which improves upon the random construction, and relies on the study of independent sets in certain Cayley graphs of the permutation group.

Circuit Complexity of Bounded Planar Cutwidth Graph Matching

- MathematicsElectron. Colloquium Comput. Complex.
- 2018

This paper disproves the conjecture that perfect matching in bounded planar cutwidth bipartite graphs is in ACC by showing that the problem is not in AC$^0[p^{\alpha}]$ for every prime $p$.

Planar Graph Perfect Matching Is in NC

- Computer Science2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
- 2018

This paper gives an NC algorithm for finding a perfect matching in a planar graph at which many new conditions, involving constraints of the polytope, are simultaneously satisfied.

Nearly Optimal Communication and Query Complexity of Bipartite Matching

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- 2022

The algorithms and lower bounds follow from simple applications of known techniques such as cutting planes methods and set disjointness and solve general linear program in the multiparty model of communication.

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