Bertsimas and Weismantel 1.19, pg 32 Given an undirected graph G = (V;E), with jV j = n and jEj = m, form a directed graph G = (V;A) by replacing…

Bertsimas and Weismantel 1.19, pg 32Given an undirected graph G = (V;E), with jV j = n and jEj = m, form a directed graph G = (V;A)by replacing each edge fi; jg in E by arcs (j; i) in A. We select a node r in V as the root node. Letyi;j = 1 if the tree contains arc (i; j) when we root the tree at node r (in other words, the solution willbe a tree with directed edges away from the root.) Let +(S) be the set of arcs going out of S and letE(S) = f(i; j) 2 Ej i; j 2 Sg. LetPdcut = fx 2 Rmj 0 x e; xe = yij + yji; 8e 2 E;Pe2A ye = n ô€€€ 1;P+(S) ye 1; r 2 S; 8S V; ye 0gandPsub = fx 2 Rmj 0 xe 1;Pe2E xe = n ô€€€ 1;Pe2E(S) xe jSj ô€€€ 1; S V; S 6= ;; V gProve that Pdcut = Psub.

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