NC algorithms for perfect matching in bipartite graphs

Establish whether there exists a deterministic NC algorithm for the perfect matching problem in bipartite graphs, for any of the decision, search, or weighted variants.

Background

The perfect matching problem is solvable in polynomial time, and randomized parallel (RNC) algorithms are known. Quasi-NC algorithms are also known for bipartite and general graphs, but a deterministic NC algorithm remains elusive.

The authors emphasize that despite significant progress, obtaining an NC algorithm for any of the standard versions (decision/search/weighted) of perfect matching—even restricted to bipartite graphs—remains a central open question in parallel complexity.