On the complexity of the spaced seeds

Optimal spaced seeds were introduced by the theoretical computer science community to bioinformatics to effectively increase homology search sensitivity. These seeds are serving many homology queries daily. However the computational complexity of finding the optimal spaced seeds remains to be open. In this paper, we prove that computing hit probability of a spaced seed in a uniform homology region is NP-hard, but it admits a probabilistic PTAS. We also show that the asymptotic hit probability is computable in exponential time in seed length, independent of the homologous region length.