final2 - COS 521 Advanced Algorithm Design Final Due Mon...

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COS 521: Advanced Algorithm Design Final Due: Mon, May 17 (5 pm) Instructions: There are a total of 6 questions on the final. No collaboration is allowed. You may refer to any reference material; however you should not explictly search for solutions to the questions on the exam. If you do refer to any sources, indicate this on your homework. 1. Given a directed graph G = ( V, E ), let f ( G ) be the minimum number of edges of G that need to be deleted so that the remaining graph does not contain any directed cycle. (i) Let D ( G ) be a collection of disjoint directed cycles in G . Show that |D ( G ) | ≤ f ( G ). (Note: |D ( G ) | is the number of cycles in D ( G )). (ii) Consider a fractional version of the previous bound. Let D ( G ) be a collection of disjoint directed cycles in G , where each cycle C ∈ D ( G ) is associated with a non- negative weight w C . Further for every edge e , the sum of weights of all cycles containing e is at most 1. Let W = w C be the sum of weights of cycles in D ( G ). Show that W f ( G ).
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