comp360_win2006_sol

# comp360_win2006_sol - Solutions to the COMP-360 Winter 2006...

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Solutions to the COMP-360 Winter 2006 Midterm 1. Selling cars (10 points) (a) The objects to be chosen are customer-car pairs. For example, selling Car B to Curly would be represented by the pair (Curly, Car B). There are thus N 2 objects to choose from. The weight of such a pair, of course, is the money earned by the salesman. Most students gave one of two possible definitions for an admissible solution. Version 1 is: An admissible solution is any set of objects such that no person and no car appears more than once. Version 2 is: An admissible solution is a set of precisely N objects such that no person and no car appears more than once. (In other words, only complete matchings between people and cars are allowed as solutions. This is reasonable because a solution of size less than N can not be optimal, assuming strictly positive car costs.) (b) The two main properties are Heredity, or Inheritance, and the Exchange Property. The Heredity property states that if S is an admissible solution and S S , then S is an admissible solution. The Exchange Property states that if S and S are two admissible solutions and | S | < | S | , then there is some object x S such that S ∪ { x } is an admissible solution. (c) Version 1 satisfies the inheritance property trivially. It does not satisfy Exchange. For example, let S = { (Larry, Car A), (Moe, Car B) } and S = { (Moe, Car A) } . We have | S | < | S | , but neither object from S can be added to S . Version 2 does not satisfy inheritance, because any strict subset of an admissible solution is not of size N

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comp360_win2006_sol - Solutions to the COMP-360 Winter 2006...

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