hw2_sol_vsr - Homework 2 Problems 3,4 and 6 Sudheer Vakati...

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Homework 2: Problems 3,4 and 6 Sudheer Vakati September 25, 2009 Problem 3 a)Formulate the problem as a circulation problem to find the feasible flow f . The miminum required flow l on each edge translates to a demand on the nodes. Solving the circulation problem gives a flow which satisfies the minimum flow value requirement on each edge. Its possible the flow in itself is not minimum. To find the minimum possible flow, build a graph G’ with the same vertex set as G. For every edge e = ( u, v) in G, add an edge ( u, v) with capacity c e - f(e) and another edge ( v,u ) with capacity f(e) - l e . Find the maximim flow f t from t to s in graph G’. The minimum flow f m ( e ) for every edge e(u,v) =f(e) - f t ( v,u ) + f t ( u,v ) . It can be seen easily that the value f m will be minimum. The pushback from t to s in G’ will reduce the flow by the maximium possible on every edge. b) Consider any s-t cut C(A,B) in G. The flow value across the cut v(C) = sum of flow on all edges out of A - Sum of flow on all edges into A. Σ e out of A l e Σ e into A f m ( e ) Σ eout of A l e Σ e into A c ( e ) — (1) Build the Graph G’ (as mentioned in part a) using flow f m . Consider the set of all vertices B reachable from t. Consider the cut C’(V-B, B) in graph G.
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