# Modeling the multiplication of two binary decision variables in undirected graph in python

Operations Research Asked by Amedeo on January 18, 2021

In an undirected graph, I’m trying to model a constraint that forcing the optimizer to set an edge $$(u,v)$$ between two nodes to only exist (= $$1$$) if the two nodes have been selected to be $$1$$. The three decision variables can be something like that:

$$z(u,v) geq y_u x_v$$ , $$quad x,y,z in {0,1}$$

and to linearize the multiplication here, I’m introducing a new decision variable $$r=xy$$ and these constraints has been added:

$$z(u,v) geq r(u,v)$$

$$r(u,v) leq y_u$$

$$r(u,v) leq x_v$$

$$r(u,v) geq y_u + x_v -1$$

Keeping in mind that this is for undirected graph where $$r(u,v)$$ is different from $$r(v,u)$$. How can I model this in Python using Pulp and NetworkX ?

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