  Permutation invariant training (PIT) is a widely used training criterion for
neural network-based source separation, used for both utterance-level
separation with utterance-level PIT (uPIT) and separation of long recordings
with the recently proposed Graph-PIT. When implemented naively, both suffer
from an exponential complexity in the number of utterances to separate,
rendering them unusable for large numbers of speakers or long realistic
recordings. We present a decomposition of the PIT criterion into the
computation of a matrix and a strictly monotonously increasing function so that
the permutation or assignment problem can be solved efficiently with several
search algorithms. The Hungarian algorithm can be used for uPIT and we
introduce various algorithms for the Graph-PIT assignment problem to reduce the
complexity to be polynomial in the number of utterances.
