  We consider a pair of identical fermions with a short-range attractive
interaction on a finite lattice cluster in the presence of strong site
disorder. This toy model imitates a low density regime of the strongly
disordered Hubbard model. In contrast to spinful fermions, which can
simultaneously occupy a site with a minimal energy and thus always form a bound
state resistant to disorder, for the identical fermions the probability of
pairing on neighboring sites depends on the relation between the interaction
and the disorder. The complexity of `brute-force' calculations (both analytical
and numerical) of this probability grows rapidly with the number of sites even
for the simplest cluster geometry in the form of a closed chain. Remarkably,
this problem is related to an old mathematical task of computing the volume of
a polyhedron, known as NP-hard. However, we have found that the problem in the
chain geometry can be exactly solved by the transfer matrix method. Using this
approach we have calculated the pairing probability in the long chain for an
arbitrary relation between the interaction and the disorder strengths and
completely described the crossover between the regimes of coupled and separated
fermions.
