The Tree Structure of Optimal Traffic Plans and their Approximation

Abstract

We shall get in this chapter a deeper insight of the structure of optimal traffic plans. Section 7.1 proves a single path property for optimal traffic plans: almost all fibers passing from x to y follow the same path between x and y. By a slight modification of the optimal traffic plan, this statement can be made strict: All fibers (not just almost all) passing by x and y follow the same path. Section 7.2 extends the single path property in the case of optimal traffic plans for the irrigation problem. In that case optimal traffic plans have no circuits, in other terms are trees. Section 7.3 uses the single path property to show that optimal traffic plans can be monotonically approximated by finite irrigation graphs. The steps leading to this conclusion have their own interest. In particular it is shown that any optimal traffic plan is a finite or countable union of trunk trees, namely trees whose fibers pass all by some point. The presentation here is inspired from [13]. Several techniques come from papers by Xia [94] and Maddalena-Solimini [58]. The proof of the bi-Lipschitz regularity of fibers with positive flow follows [95] and the pruning and theorem is borrowed from Devillanova and Solimini [79]. The monotone approximation theorem has also been proved in [58].