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.
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Fonlupt, J., Skoda, A. (2009). Strongly Polynomial Algorithm for the Intersection of a Line with a Polymatroid. In: Cook, W., Lovász, L., Vygen, J. (eds) Research Trends in Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76796-1_5
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