AMS/MBA 546 (Fall, 2010)Estie ArkinNetwork Flows: Solution sketch to homework set # 31). Recall theweighted max acyclic subgraph problem: Given a directed graphD= (N,A), withweights on the arcswij≥0, find a subset of the arcsa′⊂Asuch thatD′= (N,A′) is acyclic, and the weightofA′,w(A′), is as large as possible. Consider the following greedy algorithm: Sort the arcs by weight, fromlargest to smallest, consider the arcs in this order, and insert an arc intoA′as long as it does not create adirected cycle with the arcs already inA′. (At the startA′is empty.) Clearly this algorithm yields a feasiblesolution, however, the weight ofA′can be much less thanopt, the weight of an optimal solution. Describea family of directed graphs for whichw(A′)/opt≤c/n, wherecis some constant (that does not depend onn) andnis the number of nodes in your graph.Consider the following directed graph on nodes 1,2,...,nwith arcs (i,i+ 1), for everyi, of cost 2, and arcs(i,j) for everyi > jof cost 1. Our greedy algorithm will first pick all arcs of cost 2, and then cannot put
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