On computing the determinant in small parallel time using a small number of processors
Information Processing Letters
A taxonomy of problems with fast parallel algorithms
Information and Control
NC algorithms for computing the number of perfect matchings in K3,3-free graph and related problems
Information and Computation
On the parallel complexity of Hamiltonian cycle and matching problem on dense graphs
Journal of Algorithms
Why is Boolean complexity theory difficult?
Poceedings of the London Mathematical Society symposium on Boolean function complexity
Fast parallel algorithms for graph matching problems
Fast parallel algorithms for graph matching problems
The complexity of matrix rank and feasible systems of linear equations
Computational Complexity
Isolation, matching and counting uniform and nonuniform upper bounds
Journal of Computer and System Sciences
Matching is as easy as matrix inversion
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
The Matching Problem for Strongly Chordal Graphs is in $NC$
The Matching Problem for Strongly Chordal Graphs is in $NC$
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 combinatorial algorithm for Pfaffians
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
On the bipartite unique perfect matching problem
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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The perfect matching problem is known to be in P, in randomized NC, and it is hard for NL. Whether the perfect matching problem is in NC is one of the most prominent open questions in complexity theory regarding parallel computations. Grigoriev and Karpinski [GK87] studied the perfect matching problem for bipartite graphs with polynomially bounded permanent. They showed that for such bipartite graphs the problem of deciding the existence of a perfect matchings is in NC2, and counting and enumerating all perfect matchings is in NC3. For general graphs with a polynomially bounded number of perfect matchings, they show both problems to be in NC3. In this paper we extend and improve these results. We show that for any graph that has a polynomially bounded number of perfect matchings, we can construct all perfect matchings in NC2. We extend the result to weighted graphs.