Tight integral duality gap in the Chinese Postman problem

Let G = (V, E) be a graph and let w be a weight function w:E →Z⁺. Let {Mathematical expression} be an even subset of the vertices of G. A T-cut is an edge-cutset of the graph which divides T into two odd sets. A T-join is a minimal subset of edges that meets every T-cut (a generalization of solutions to the Chinese Postman problem). The main theorem of this paper gives a tight upper bound on the difference between the minimum weight T-join and the maximum weight integral packing of T-cuts. This difference is called the (T-join) integral duality gap. Let τ_w be the minimum weight of a T-join, and let v_w be the maximum weight of an integral packing of T-cuts. If F is a non-empty minimum weight T-join, and n_F is the number of components of F, then we prove that τ_w-v_w≤n_F-1. This result unifies and generalizes Fulkerson's result for |T|=2 and Seymour's result for |T|= 4. For a certain integral multicommodity flow problem in the plane, which was recently proved to be NP-complete, the above result gives a solution such that for every commodity the flow is less than the demand by at most one unit.