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Unformatted text preview: Computer Science C73 November 19, 2007 Scarborough Campus University of Toronto Homework Assignment #4 Due: December 3, 2007, by 12 noon (in the course drop box) Appended to this document is a cover page for your assignment. Fill it out, staple your answers to it, and deposit the resulting document into the course drop box. Please do not enclose your assignment in an envelope. Question 1. (10 marks) a. (2 marks) Give an example of a flow graph G with integer capacities, and a maximum flow f in G such that for some edge e , f ( e ) is not an integer. Prove that your flow f is a maximum flow in G . b. (5 marks) Let G be any flow graph with integer capacities, and f be any maximum flow in G . Assume that the value of f is positive (i.e., the source and sink are connected through at least one path in which every edge has positive capacity). Prove that there is an edge e such that f ( e ) is a positive integer. (In other words, it is not possible to construct an example as in part (a) where every edge used by the flow carries a fractional amount of traffic.) c. (3 marks) Let G be any flow graph with integer capacities, and m be the value of the maximum flow in G . Prove that, for each integer k such that 0 ≤ k ≤ m , there is a flow f that has value k . Question 2. (20 marks) The Ford-Fulkerson algorithm leaves the choice of augmenting path unspecified....
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