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week3 - 4 The project selection problem One of the...

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4 The project selection problem One of the surprising aspects of the maximum-flow minimum-cut theorem is that while we started thinking about solving one optimization problem, we ended up solving two problems for the price of one. We now also have the possibility of, given a directed graph G = ( N, E ), with a specified source node s and sink node t , where each directed edge has an associated capacity, to find the cut of minimum capacity. We next give a new optimization problem that can be formulated as a minimum-cut problem. The project selection problem is as follows. There is a collection of projects that one might take on: P = { P 1 , P 2 , . . . , P m } . Each project P i has an associated benefit b i , i = 1 , . . . , m . There is also a collection of tools T = { 1 , . . . , n } that are needed for doing these projects. Each tool j has an associated cost c j , j = 1 , . . . , n . (One might imagine that the projects are clients that a management consulting firm might take on, and the tools are software licenses that are needed for some of these projects.) For each project P i , there is a subset of tools T i ⊆ T that are required to do that project; once the tool is purchased, it may be used for as many different projects as required. The aim is to select a subset of the project, and purchase the requisite tools so as to maximize the net profit associated with these activ- ities (where the net profit is the total benefit accrued minus the total cost incurred). We will show how to formulate the project selection problem as a min- imum cut problem. A priori, this seems extremely surprising since there is no graph as part of the description of the problem, no apparent cuts, and even we trying to maximize something on the one hand, whereas the min-cut problem is a minimization problem. We will do this formulation by the same 3-step process we have repeated for other problems. First, we will show how to map a project selection input into an input for the minimum cut prob- lem. Then we will demonstrate that feasible solutions to the project selection problem have a precise correspondence to feasible solutions for the resulting minimum cut problem. Finally, we will show that optimizing the objective of the minimum cut problem exactly corresponds to finding the best selec- tion of projects. An example of an input to the project selection problem is as follows. Suppose that the set of tools is { A, B, C, D } , and suppose that the set of projects is { P 1 , P 2 , P 3 } . The requirements for the projects are as follows: the first requires tools A , B , and C ; the second requires B and C ; and the last requires C and D . The tool costs are, respectively, 5, 10, 5, and 28
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15; the project benefits are, respectively, 15, 20, and 10. At first, it seems that there is no graph structure here at all. However, it is extremely natural to represent the requirements as a graph, in a way reminiscent of what we did for the assignment problem earlier. Suppose that we construct a graph with two sets of nodes: one set corresponding to the projects, and one set corresponding to the tools. We can draw the first set on
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