We are given finitely many polyhedra defined by linear constraints, using the same
constraint matrix and different right-hand sides. We consider a simple constraint system and
give necessary and sufficient conditions for this system to define the union of the polyhedra.
We also show that deciding whether the system does define the union is NP-hard.

Jeroslow [l] raises the question of when Q= T. The motivation is that T is defined using linear
constraints with the auxiliary variables Xh. Thus, when Q= T, the problem of maximizing a
linear objectiveover the union of Ph can be done by solving a linear program of modest size.
In particular, it is not necessary to make one copy of A for each h.[l] gives a sufficient
condition (Theorem 1 below) for Q= T. In this note we give a modification which is simpler
and includes more cases (Theorem 2). Then we give a weaker sufficient condition (Theorem …