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hw7 - AMS/MBA 546(Fall 2010 Estie Arkin Network Flows...

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AMS/MBA 546 (Fall, 2010) Estie Arkin Network Flows Homework Set # 7 Due Tuesday, November 9, 2010. Suggested reading Chapters 6,7 of [AMO]. 1). [6.47] In a directed acyclic network G , certain arcs are coloured blue. Consider the problem of covering the blue arcs by directed paths, which can start and end at any node (these paths may contain blue arcs and non blue arcs). Show that the minimum number of directed paths needed to cover the blue arcs is equal to the maximum number of blue arcs that satisfy the property that no two of these arcs belong to the same path. Will this result be valid if G contains directed cycles? (Hint: use the min flow max cut theorem.) This is a theorem due to Dilworth. It is sometimes phrased using the following definitions: A partial order is a relation between elements a 1 ,a 2 ,...,a n is such that it is transitive, asymetric (i.e., if a i a j then a j negationslash≺ a i ), and for no i a i a i . If either a i a j or a j a i we say that the two elements are comparable.
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