Multiple paternity: determining the minimum number of sires of a large brood

We describe an efficient algorithm for determining exactly the minimum number of sires consistent with the multi-locus genotypes of a mother and her progeny. We consider cases where a simple exhaustive search through all possible sets of sires is impossible in practice because it would take too long to complete. Our algorithm for solving this combinatorial optimization problem avoids visiting large parts of search space that would not result in a solution with fewer sires. This improvement is of particular importance when the number of allelic types in the progeny array is large and when the minimum number of sires is expected to be large. Precisely in such cases, it is important to know the minimum number of sires: this number gives an exact bound on the most likely number of sires estimated by a random search algorithm in a parameter region where it may be difficult to determine whether it has converged. We apply our algorithm to data from the marine snail, Littorina saxatilis.