Strongly Polynomial Algorithm for the Intersection of a Line with a Polymatroid

We present a new algorithm for the problem of determining the intersection of a half-line \(\Delta_{u}=\{x\in \mathbb{R}^{N}\:|\:x=\lambda u\;\mathrm {for}\;\lambda \geq 0\}\) with a polymatroid. We then propose a second algorithm which generalizes the first algorithm and solves a parametric linear program. We prove that these two algorithms are strongly polynomial and that their running time is O(n 8+γ n 7) where γ is the time for an oracle call. The second algorithm gives a polynomial algorithm to solve the submodular function minimization problem and to compute simultaneously the strength of a network with complexity bound O(n 8+γ n 7).