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. 

Figures from this paper

Guest Column: Parallel Algorithms for Perfect Matching
TLDR
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.
Linear matroid intersection is in quasi-NC
TLDR
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.
A deterministic parallel algorithm for bipartite perfect matching
TLDR
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.
Perfect Bipartite Matching in Pseudo-Deterministic RNC
TLDR
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
TLDR
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.
Labeling the complete bipartite graph with no zero cycles
TLDR
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
TLDR
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
  • Nima AnariV. Vazirani
  • Computer Science
    2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2018
TLDR
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
TLDR
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.
...
...

References

SHOWING 1-10 OF 68 REFERENCES
The Polynomially Bounded Perfect Matching Problem Is in NC 2
TLDR
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.
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
Linear matroid intersection is in quasi-NC
TLDR
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.
NC Algorithms for Computing the Number of Perfect Matchings in K3, 3-free Graphs and Related Problems
TLDR
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.
Perfect Bipartite Matching in Pseudo-Deterministic RNC
TLDR
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).
Bipartite Perfect Matching in Pseudo-Deterministic NC
TLDR
It is proved that resolving the decision question NC = RNC, would imply an NC algorithm for finding a bipartite perfect matching and finding a DFS tree in NC.
Exact Perfect Matching in Complete Graphs
TLDR
It is shown that for complete and bipartite complete graphs, the exactperfect matching problem is logspace equivalent to the perfect matching problem, which means an efficient parallel algorithm for perfect matching would carry over to the exact perfect match problem for this class of graphs.
Randomized parallel algorithms for matroid union and intersection, with applications to arboresences and edge-disjoint spanning trees
The strong link between matroids and matching is used to extend the ideas that resulted in the design of Random NC algorithms for matching to obtain RNC algorithms for the well-known problems of
A new NC-algorithm for finding a perfect matching in bipartite planar and small genus graphs (extended abstract)
TLDR
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.
...
...