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

Summary

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).

This research was supported by a grant of “France Telecom R&D Sophia Antipolis” during three years.