  We prove new necessary and sufficient conditions to carry out a compact
linearization approach for a general class of binary quadratic problems subject
to assignment constraints as it has been proposed by Liberti in 2007. The new
conditions resolve inconsistencies that can occur when the original method is
used. We also present a mixed-integer linear program to compute a
minimally-sized linearization. When all the assignment constraints have
non-overlapping variable support, this program is shown to have a totally
unimodular constraint matrix. Finally, we give a polynomial-time combinatorial
algorithm that is exact in this case and can still be used as a heuristic
otherwise.
