In this paper we study a set of problems that are of considerable importance to Statistical Machine Translation (SMT) but which have not been addressed satisfactorily by the SMT research community. Over the last decade, a variety of SMT algorithms have been built and empirically tested whereas little is known about the computational complexity of some of the fundamental problems of SMT. Our work aims at providing useful insights into the the computational complexity of those problems. We prove that while IBM Models 1-2 are conceptually and computationally simple, computations involving the higher (and more useful) models are hard
