Fig. 7

Lemma 6 proves that any solution \((S,\psi )\) that is optimal on sub-trees rooted at Z in \(\Gamma \) must also be optimal (among all solutions with \(\psi \)’s behavior on \(\bigcup _{y\in Y}\hbox {YW}_{{y}}^{\Gamma }\) (gray box on top)) on all sub-trees of \(\Gamma \) that are rooted below Z (at Y). That is, no solution \((S^y,\psi _y)\) can be better than \((S,\psi )\) on the sub-network induced by \(\Gamma _y\) for any \(y\in Y\). To prove this, a new solution \((S^*,\psi ^*)\) is constructed by replacing the sub-solution of \((S,\psi )\) below Y by the sub-solutions \((S^y,\psi _y)\) below Y