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sol7 - AMS/MBA 546(Fall 2010 Estie Arkin Network Flows...

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AMS/MBA 546 (Fall, 2010) Estie Arkin Network Flows: Solution sketch to homework set # 7 1). [6.47] Let B be any subset of blue arcs with the property that no two arcs in B can be in any directed path. Note that the number of paths needed to cover all blue arcs must be at least | B | . Thus it suffices to show that there is some set B and some collection of paths P covering all blue arcs, such that | B | = | P | . Formulate the problem of covering paths as minimum flow: Add a sources node s and a sink node t with arcs ( s,i ) and ( i,t ) for each node i . For each blue arc, let the lower bound be 1. All other lower bounds are zero, and all upper bounds are infinity. The minimum value of flow satisfying these bounds is equal to the minimum number of paths needed to cover all blue arcs. Simply think of each path as an augmenting path with s added at the beginning and t at the end of the directed path. The min flow is equal to max( i S,j/ S l ij - i/ S,j S u ij ). Since all upper bounds are infinity, we can consider only cuts with no arcs directed from ¯ S to S . Since such arcs do not exists, any two arcs in the cut (which must be from S to ¯ S ) cannot be in the same directed path. Thus the min flow is equal to the number of blue arcs in the cut (since other arcs have lower bound zero).
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