Dr. Rasi Flosi-Starkasi has prepared 50 problems for the exam of his module “Advanced Algorithmic…

Dr. Rasi Flosi-Starkasi has prepared 50 problems for the exam of his module “Advanced Algorithmic Techniques”. Each one of these problems has two attributes: – Its type: it is either a problem on graph algorithms, approximation algorithms or randomised algorithms. – Its difficulty: it is either easy, moderate or difficult. For example, it could be that Problem #34 is an easy problem on approximation algorithms. Dr. Flosi-Starkasi would like to prepare an exam consisting of 24 of those problems, but he wants to make sure that the exam containts 8 problems on graph algorithms, 8 problems on approximation algorithms and 8 problems on ran- domised algorithms and at the same time 8 easy problems, 8 moderate problems and 8 difficult problems. Model this problem as a maximum flow problem, by explaining all the parameters of the flow network. Explain how to find a feasible exam set (i.e., satisfying the constraints set by Dr. Flosi-Starkasi above) from the maximum flow in the network, if it exists, or how to decide that it does not exist. [15 marks) It turned out that the exam set by Dr. Flosi-Starkasi in the previous part of the problem was really boring. For that reason, he decided to record an additional attribute for each problem, its entertainment value, which is a real number 2 between 0 and 1. Dr. Flosi-Starkasi would now like to find a feasible exam (satisfying the constraints set in the previous part) which maximises the total entertainment value (i.e., the sum of the entertainment values of the problems included in the exam). Model this problem and an integer linear program (ILP). Explain the variables and the constraints of your ILP. [10 marks)
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