Tight integral duality gap in the Chinese Postman problem

AbstractLetG = (V, E) be a graph and letw be a weight functionw:E →Z+. Let $$T \subseteq V$$ be an even subset of the vertices ofG. AT-cut is an edge-cutset of the graph which dividesT into two odd sets. AT-join is a minimal subset of edges that meets everyT-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 weightT-join and the maximum weight integral packing ofT-cuts. This difference is called the (T-join) integral duality gap. Letτw be the minimum weight of aT-join, and letvw be the maximum weight of an integral packing ofT-cuts. IfF is a non-empty minimum weightT-join, andnF is the number of components ofF, then we prove thatτw—vw≤nF−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.