  Given a combinatorial optimization problem, we aim at characterizing the set
of all instances for which every feasible solution has the same objective
value.
  Our central result deals with multi-dimensional assignment problems. We show
that for the axial and for the planar $d$-dimensional assignment problem
instances with constant objective value property are characterized by
sum-decomposable arrays. We provide a counterexample to show that the result
does not carry over to general $d$-dimensional assignment problems.
  Our result for the axial $d$-dimensional assignment problem can be shown to
carry over to the axial $d$-dimensional transportation problem. Moreover, we
obtain characterizations when the constant objective value property holds for
the minimum spanning tree problem, the shortest path problem and the minimum
weight maximum cardinality matching problem.
