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

In: Research Trends in Combinatorial Optimization

Author

Listed:
  • Jean Fonlupt

    (CNRS et Université Pierre et Marie Curie (Paris 6), Equipe Combinatoire et Optimisation)

  • Alexandre Skoda

    (CNRS et Université Pierre et Marie Curie (Paris 6), Equipe Combinatoire et Optimisation)

Abstract

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).

Suggested Citation

  • Jean Fonlupt & Alexandre Skoda, 2009. "Strongly Polynomial Algorithm for the Intersection of a Line with a Polymatroid," Springer Books, in: William Cook & László Lovász & Jens Vygen (ed.), Research Trends in Combinatorial Optimization, chapter 5, pages 69-85, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-76796-1_5
    DOI: 10.1007/978-3-540-76796-1_5
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