  In this note, we show that the linear programming for computing the
quasi-additive bound of the formula size of a Boolean function presented by
Ueno [MFCS'10] is equivalent to the dual problem of the linear programming
relaxation of an integer programming for computing the protocol partition
number. Together with the result of Ueno [MFCS'10], our results imply that
there exists no gap between our integer programming for computing the protocol
partition number and its linear programming relaxation.
