  This paper addresses the problem of enumerating all supported efficient
solutions for a linear multi-objective integer minimum cost flow problem
(MOIMCF).
  First, we highlight an inconsistency in various definitions of supported
nondominated vectors for multi-objective integer linear programs (MOILP).
Several characterizations for supported nondominated vectors/efficient
solutions are used in the literature, which are equivalent in the non-integer
case. However, they may lead to different sets of supported nondominated
vectors/efficient solutions for MOILPs. This motivates us to summarize
equivalent definitions and characterizations for supported efficient solutions
and to distinguish between supported and weakly supported efficient solutions.
  In this paper we derive an output-polynomial time algorithm to determine all
supported efficient solutions for MOIMCF problems.
  This is the first approach that solves this general problem in
output-polynomial time.
  Moreover, we prove that the existence of an output-polynomial time algorithm
to determine all weakly supported nondominated vectors (or all weakly supported
efficient solutions) for a MOIMCF problem with a fixed number of d>3 objectives
can be excluded, unless P = NP.
