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(x_1,\ldots,x_n)=x_{n+1} such as conjunction, disjunction, equivalence, and implication using n+1 boolean variables x1,...,x_{n+1}. For the case that the value f(x1, ...,xn) is not needed for further computations, we even give 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.
Highlights
Linear programming is a powerful tool, studied for over 50 years, that can be used to define a lot of very important optimization problems [3, 6]
We extend the definition of the binary boolean functions and and nor given in [7] and of the functions and, or, and not given in [2] and [4] to all possible 16 binary boolean functions
Our first two given formulations imply that every binary boolean function can be defined by linear programming constraints, we suggest the following conditions in order to get more efficient formulations and to prove
Summary
Linear programming is a powerful tool, studied for over 50 years, that can be used to define a lot of very important optimization problems [3, 6]. The linear programming problem is, given an objective function and a finite set of constraints, to find an optimal solution. A linear program can be expressed as a task of minimizing cT x subject to the constraints Ax ≥ b and x ≥ 0. In integer linear programming problems (IP), the variables are all required to be integers and in binary linear programming problems (BIP), each variable can only take the two values, zero or one. The task to find equivalent linear programming formulations for optimization problems is often challenging, see e.g. In this paper we give efficient binary linear programming formulations (BIP formulations) for several important boolean functions such as n-ary conjunction, n-ary disjunction, equivalence, and implication.
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