Sol_prelim2 - Solution Set for CS 482 Prelim 2 April 8 2008 Questions in red solutions in black PROBLEM 1(20 points PART A(15 points Find a maximum

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Solution Set for CS 482, Prelim 2 April 8, 2008 Questions in red, solutions in black. PROBLEM 1 (20 points) PART A (15 points) Find a maximum flow and minimum s-t cut in the flow network G shown here. The source and sink are s and t, respectively. The edge capacities are denoted by numbers alongside the edges in the diagram. You may express the minimum s-t cut by writing a partition of the vertex set {s,v,w,x,y,t} into two sets (A,B). A = {s,w,x,y} B = {v,t} The flow shown at right is one of the three integer maximum flows in this network. Any flow of value 21 constitutes a correct solution. s w y t v x 10 8 6 2 1 3 2 10 8 9 s w y t v x 9 9 0 8 8 10 2 2 4 0
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PART B (5 points) Give a counterexample to the following statement. Let G be a flow network with source s and sink t. Suppose f is a maximum flow in G and (A,B) is a minimum s-t cut in G. If the capacity of every edge of G is increased by 1, then the maximum flow value increases from v(f) to v(f)+k, where k is the number of edges from A to B, i.e. the number of edges (u,v) with u in A and v in B. Your counterexample should consist of an explicit flow network G that violates the statement. You should describe G (the vertices, edges, source and sink nodes, and edge capacities) in words, in pictures, or both. If you describe G using a diagram, make sure that all vertices and edges of G can be clearly identified in the diagram. There are many counterexamples. Here is one. In this flow network the maximum flow value is v(f)=3, and the unique minimum s-t cut is (A,B), where A={s} and B={w,x,y,z,t}. Hence k=3, and v(f)+k=6. If we increase the capacity of each edge by 1, we obtain a new flow network (shown in the second diagram) whose maximum flow value is 5, not 6. y t 4 2 2 1 w 1 z 1 x s y t 5 3 3 2 w 2 3 z 2 x s 2
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PROBLEM 2 (20 points) Ford-Fulkerson University has many university-wide committees that need to be filled with professors. The university has n professors and m committees. Each committee k has a required number of members, r k . No professor is allowed to serve on more than c committees. For each professor j, there is a list L j of committees on which he or she is qualified to serve. (a) [12 points] Design a polynomial-time algorithm to determine whether it is possible to fill each committee with the required number of qualified professors. (b) [8 points] Ford-Fulkerson University is organized into d departments, and each professor belongs to only one department. The university institutes a new rule that no committee is allowed to have more than one professor from the same department. Design a polynomial-time algorithm to determine whether it is possible to fill each committee with the required number of qualified professors from different departments. Note:
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This note was uploaded on 10/02/2008 for the course CS 482 taught by Professor Kleinberg during the Spring '08 term at Cornell University (Engineering School).

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Sol_prelim2 - Solution Set for CS 482 Prelim 2 April 8 2008 Questions in red solutions in black PROBLEM 1(20 points PART A(15 points Find a maximum

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