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sol3 - 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 # 3 1). Recall the weighted max acyclic subgraph problem : Given a directed graph D = ( N,A ), with weights on the arcs w ij 0, find a subset of the arcs a A such that D = ( N,A ) is acyclic, and the weight of A , w ( A ), is as large as possible. Consider the following greedy algorithm: Sort the arcs by weight, from largest to smallest, consider the arcs in this order, and insert an arc into A as long as it does not create a directed cycle with the arcs already in A . (At the start A is empty.) Clearly this algorithm yields a feasible solution, however, the weight of A can be much less than opt , the weight of an optimal solution. Describe a family of directed graphs for which w ( A ) /opt c/n , where c is some constant (that does not depend on n ) and n is the number of nodes in your graph. Consider the following directed graph on nodes 1,2,..., n with arcs ( i,i + 1), for every i , of cost 2, and arcs ( i,j ) for every i > j of cost 1. Our greedy algorithm will first pick all arcs of cost 2, and then cannot put
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