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

TL;DR

This paper establishes a theoretical equivalence between the linear programming approach for the quasi-additive bound of Boolean function complexity and the protocol partition number, showing no gap exists between their integer and relaxed forms.

Contribution

It reveals the equivalence between two linear programming formulations related to Boolean function complexity measures, unifying previous approaches.

Findings

Linear programming for quasi-additive bounds is equivalent to the dual of the relaxation for protocol partition number.

No gap exists between the integer programming and its linear relaxation for the protocol partition number.

The results unify different complexity measures of Boolean functions.

Abstract

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.