Strongly polynomial algorithm for the intersection of a line with a polymatroid

Abstract

We present a new algorithm for the problem of determining the intersection of a half-line δu = (x∈ℝN|x =λu for λ ≥ 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(n8+γ n7) 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(n8+γ n 7). © 2009 Springer Berlin Heidelberg.