The Complexity of Computing the Permanent

where the summation is over the n! permutations of (1,2, . . . , n). It is the same as the determinant except that all the terms have positive sign. Despite this similarity, while there are efficient algorithms for computing the determinant all known methods for evaluating the permanent take exponential time. This discrepancy is annoyingly obvious even for small matrices, and has been noted repeatedly in the literature since the last century [IS]. Several attempts have been made to determine whether the permanent could be reduced to the determinant via some simple matrix transformation. The results have always been negative, except in certain special cases [12,X3,16). The aim of this paper is to explain the apparent intractability of the permanent by showing that it is “complete” as far as counting problems. The results can be summarized informally as follows: