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

Jean Fonlupt (1) , Alexandre Skoda (2, 1, 3)

1 C&O - Equipe combinatoire et optimisation
2 CES - Centre d'économie de la Sorbonne
3 France Telecom R&D [Sophia-Antipolis]

We present a new algorithm for the problem of determining the intersection of a half-line 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 ).

Mots clés: Algorithm; graph; strength of a graph; submodular function; matroid; polymatroid; parametric linear programming

Domaines: Recherche opérationnelle [math.OC]; Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Jean Fonlupt, Alexandre Skoda. Strongly polynomial algorithm for the intersection of a line with a polymatroid. Research Trends in Combinatorial Optimization, Springer Berlin Heidelberg, pp.69-85, 2009, ⟨10.1007/978-3-540-76796-1_5⟩. ⟨hal-00634187⟩