Coflow polyhedra

Abstract

A system L of linear inequalities in the variables x is called totally dual integral (TDI) if for every linear function cx such that c is all integers, the dual of the linear program: maximize { cx≔ x satisfies L} has an integer-valued optimum solution or no optimum solution. A linear system L is called box TDI if L together with any inequalities b⩽ x⩽ a is TDI. The main result of this paper is the Coflow Polyhedron Theorem: For any digraph G with node-set V ( G), and for any fixed rational-valued d> = ( d v ≔ v ϵ V ( G)), the following system of inequalities in variables x = ( x v ≔ v ϵ V ( G)) is box TDI. ∀ dicircuit C of G, ∑{x v≔v∈C} ⩽ ∑ {d v≔v∈C}. By Edmonds and Giles' basic theorem on TDI systems, the Coflow Polyhedron Theorem provides many combinatorial min-max relations. In particular, it implies min-max theorems for the maximum weight union of k antichains and the maximum weight union of k chains in a poset. It allows us to prove that a class of graphs which generalize comparability graphs are perfect, and prove a theorem which for acyclic digraphs generalizes the Gallai–Milgram Theorem. The proof of dual integrality of the inequality system described above uses network flow techniques. The proof of primal integrality then follows by the Edmonds-Giles Theorem, or can be proved directly using the concept of projection of a polyhedron.