Constructing a perfect matching is in random NC
Combinatorica
Matching is as easy as matrix inversion
Combinatorica
Perfect matching for regular graphs is AC0-hard for the general matching problem
Journal of Computer and System Sciences
Flow in Planar Graphs with Multiple Sources and Sinks
SIAM Journal on Computing
Fast parallel algorithms for graph matching problems
Fast parallel algorithms for graph matching problems
Bipartite Edge Coloring in $O(\Delta m)$ Time
SIAM Journal on Computing
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Matching Theory (North-Holland mathematics studies)
Matching Theory (North-Holland mathematics studies)
The matching problem for bipartite graphs with polynomially bounded permanents is in NC
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
A fast parallel algorithm for routing in permutation networks
IEEE Transactions on Computers
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The perfect matching problem for general graphs reduces to the same for regular graphs. Even finding an NC algorithm for the perfect matching problem in cubic (3-regular) or 4-regular graphs will suffice to solve the general problem (see [DK 92]). For regular bipartite graphs an NC algorithm is already known [LPV 81], while [SW 96] give an NC algorithm for cubic-bipartite graphs. We present a new and conceptually simple parallel algorithm for finding a perfect matching in d-regular bipartite graphs. When d is small (polylogarithmic) our algorithm in fact runs in NC. In particular for cubic-bipartite graphs, our algorithm as well as its analysis become much simpler than the previously known algorithms for the same. Our techniques are completely different from theirs. Interestingly, our algorithm is based on a method used by [MV 00] for finding a perfect matching in planar-bipartite graphs. So, it is remarkable that, circumventing the planarity, we could still make the same approach work for a non-planar subclass of biparitite graphs.