  A list of complex numbers is realizable if it is the spectrum of a
nonnegative matrix. In 1949 Suleimanova posed the nonnegative inverse
eigenvalue problem (NIEP): the problem of determining which lists of complex
numbers are realizable. The version for reals of the NIEP (RNIEP) asks for
realizable lists of real numbers. In the literature there are many sufficient
conditions or criteria for lists of real numbers to be realizable. We will
present an unified account of these criteria. Then we will see that the
decision problem associated to the RNIEP is NP-hard and we will study the
complexity for the decision problems associated to known criteria.
