  In this note, we present a unified approach to the problem of existence of a
potential for the optimal transport problem with respect to non-traditional
cost functions, that is costs that assume infinite values. We establish a new
method which relies on proving solvability of a special (possibly infinite)
family of linear inequalities. We give a necessary and sufficient condition on
the coefficients that assure the existence of a solution, and which in the
setting of transport theory we call $c$-path-boundedness. This condition fully
characterizes sets that admit a potential and replaces $c$-cyclic monotonicity
from the classical theory, i.e. when the cost is real-valued. Our method also
gives a new and elementary proof for the classical results of Rockafellar,
Rochet and R\"uschendorf.
