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 $${\bar{\mathrm {r}}}(H_1, H_2) \le \sigma (H_1) {\bar{ \mathrm {r}}}(H_1) {\bar{\mathrm {r}}}(H_2)$$ , where $${\bar{ \mathrm{r}}} (H) := \max ( \mathrm{r} (H)-1,0)$$ is the reduced rank of H, $$\mathrm {r}(H)$$ is the rank of H, and $${\bar{\mathrm {r}}}(H_1, H_2)$$ 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 $${\bar{\mathrm {r}}}(H_1, H_2^*) = \sigma (H_1) {\bar{\mathrm {r}}}(H_1) {\bar{ \mathrm {r}}}(H_2^*)$$ , 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 $$H_1$$ .