AMS/MBA 546 (Fall, 2010)Estie ArkinNetwork FlowsHomework Set # 7Due Tuesday, November 9, 2010.Suggested reading Chapters 6,7 of [AMO].1). [6.47] In a directed acyclic networkG, certain arcs are coloured blue. Consider the problem of coveringthe blue arcs by directed paths, which can start and end at any node (these paths may contain blue arcsand non blue arcs). Show that the minimum number of directed paths needed to cover the blue arcs is equalto the maximum number of blue arcs that satisfy the property that no two of these arcs belong to the samepath. Will this result be valid ifGcontains 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 elementsa1,a2,...,anis such that it is transitive, asymetric (i.e., ifai≺ajthenajnegationslash≺ai), and for noi ai≺ai. If eitherai≺ajoraj≺aiwe say that the two elements are comparable.
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