Strong formulations for assignment and network design often split each capacitated
resource into one copy per level, the number of items it serves, and
then link the levels of different resources. We ask when linking disjunctive
blocks preserves exactness, which of their levels must be exact, and how the
answer depends on the structure of the coupling. The linked formulation is
ideal if and only if the links are integral when items, weights and capacities
are ignored, and each block is exact at the levels the links actually allow.
Sufficiency follows from disjunctive programming; necessity holds exactly for
the levels used by an admissible profile, so links can exclude a bad level but
never repair one they admit. Our main result characterizes when the formulation
is sharp, that is, exact in the original variables. Integral links are then
not needed: it suffices that they spread each resource’s level distribution, in
the convex order, around an admissible one, and this criterion is correct precisely
when the level capacities satisfy a midpoint inequality, which tightened
capacities can violate. Links are integral on forests and oriented sublattices;
for a count relation an arc-flow network describes them, and a shortest path
in it shows how cut-set inequalities designed for Steiner arborescences can
cut off optimal solutions in unsplittable network design.