More specifically, I'm looking at the problem of applying an algorithm for computing the permanent of a sparse matrix of binary entries (0s and 1s) to a matrix that has entries of positive and negative integers (but still sparsely).
I am not sure how hard this is. Clearly the permanent of a general matrix can be negative so it cannot be written as the permanent of a matrix with non-negative entries.
However, there may be some kind of way to re-represent the permanent of the general matrix in terms of other permanents.
That is, Per(A) = Per(A1) +/- Per(A2) +/- Per(A3) +/-... for some matrices A1,A2,A3, or something like that.
This is related to the problem of rerepresenting weighted graphs as non-weighted graphs through the connection of the permanent and cycle covers on the adjacency graph.