Efficient binary linear programming formulations for boolean functions

Abstract

A very useful tool when designing linear programs for optimization problems is the formulation of logical operations by linear programming constraints. We give efficient linear programming formulation of important n-ary boolean functions f(x1,..., xn) = xn+1 such as conjunction, disjunction, equivalence, and implication using n + 1 boolean variables x1,..., xn+1. For the case that the value f(x1,... xn) is not needed for further computations, we even give a more compact formulation. Our formulations show that every binary boolean function f(x1, x2) = x3 can be realized by the only three boolean variables x1,x2,x3 and at most four linear programming constraints.