  We study the intersection of finitely generated subgroups of free groups by
utilizing the method of linear programming. We prove that if $H_1$ is a
finitely generated subgroup of a free group $F$, then the WN-coefficient
$\sigma(H_1)$ of $H_1$ is rational and can be computed in deterministic
exponential time in the size of $H_1$. This coefficient $\sigma(H_1)$ is the
minimal nonnegative real number such that, for every finitely generated
subgroup $H_2$ of $F$, it is true that $\bar {\rm r}(H_1, H_2) \le \sigma(H_1)
\bar {\rm r}(H_1) \bar {\rm r}(H_2)$, where $\bar{ {\rm r}} (H) := \max ( {\rm
r} (H)-1,0)$ is the reduced rank of $H$, ${\rm r} (H)$ is the rank of $H$, and
$\bar {\rm r}(H_1, H_2)$ is the reduced rank of the generalized intersection of
$H_1$ and $H_2$. We also show the existence of a subgroup $H_2^* = H_2^*(H_1)$
of $F$ such that $\bar {\rm r}(H_1, H_2^*) = \sigma(H_1) \bar {\rm r}(H_1) \bar
{\rm r}(H_2^*)$, the Stallings graph $\Gamma(H_2^*)$ of $H_2^*$ has at most
doubly exponential size in the size of $H_1$ and $\Gamma(H_2^*)$ can be
constructed in exponential time in the size of $H_1$.
