On Multiflow Lexicographics

Abstract

Given an undirected Eulerian network with the terminal-set { s } ∪ T , we call a vector ξ = ( ξ ( t ) : t ∈ T ) feasible if there exists an integer maximum multiflow having exactly ξ ( t ) ( s , t )-paths for each t ∈ T . This paper contributes to describing the set Ξ of feasible vectors. First, the feasible vectors are shown to be bases of a polymatroid ( T , p ) forming a proper part of the polytope defined by the supply–demand conditions; p ( V ) = max { ξ ( V ) : ξ ∈ Ξ }, V ⊆ T is described by a max–min theorem. The question whether Ξ contains all the integer bases, thereby admitting a polyhedral description, remains open. Second, the lexicographically minimum (and thereby maximum) feasible vector is found, for an arbitrary ordering of T . The results are based on the integrality theorem of A. Karzanov and Y. Manoussakis (Minimum (2, r )-metrics and integer multiflows, Europ. J. Combinatorics (1996) 17 , 223–232) but we develop an original approach, also providing an alternative proof to this theorem.