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It is very easy to determine the satisfiability of a boolean formula in DNF: such a formula is satisfiable if and only if it contains a satisfiable conjunction ( one that does not contain a variable and its negation ), whereas counting the number of satisfying assignments is # P-complete.
It was known before that the decision problem " Is there a perfect matching for a given bipartite graph?
" can be solved in polynomial time, and in fact, for a graph with V vertices and E edges, it can be solved in O ( VE ) time.
The problem of counting the number of perfect matchings ( or in directed graphs: the number of vertex cycle covers ) is known to be equivalent to the problem of the computation of the permanent of a matrix.
The perfect matching counting problem was the first counting problem corresponding to an easy P problem shown to be # P-complete, in a 1979 paper by Leslie Valiant which also defined the classes # P and # P-complete for the first time.
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