A Relation between the Protocol Partition Number and the Quasi-Additive Bound

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