Linear programming and the intersection of subgroups in free groups

We study the intersection of finitely generated subgroups of free groups by utilizing the method of linear programming. We prove that if H1 is a finitely generated subgroup of a free group F , then the Walter Neumann coefficient σ(H1) of H1 is rational and can be computed in deterministic exponential time of size of H1. This coefficient σ(H1) is a minimal nonnegative real number such that, for every finitely generated subgroup H2 of F , it is true that r̄(H1,H2) ≤ σ(H1 )̄r(H1 )̄r(H2), where r̄(H) := max(r(H) − 1, 0) is the reduced rank of H, r(H) is the rank of H, and r̄(H1, H2) is the reduced rank of a generalized intersection of H1,H2.