ps8sol

# ps8sol - Introduction to Algorithms Massachusetts Institute...

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Introduction to Algorithms December 1, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Piotr Indyk and Charles E. Leiserson Handout 31 Problem Set 8 Solutions Problem 8-1. Inspirational ﬁres To foster a spirit of community and cut down on the cliqueishness of various houses, MIT has decided to sponsor community-building activities to bring together residents of different living groups. Speciﬁcally, they have started to sponsor ofﬁcial gatherings in which they will light copies of CLRS on ﬁre. Let G be the set of living groups at MIT, and for each g G , let residents ( g ) denote the number of residents of living group g . President Hockﬁeld has asked you to help her out with the beginning of her administration. She gives you a list of book-burning parties P that are scheduled for Friday night. For each party p P , you are given the number size ( p ) of people who can ﬁt into the site of party p . The administration’s goal is to issue party invitations to students so that no two students from the same living group receive invitations to the same book-burning party. Formally, they want to send invitations to as many students as possible while satisfying the following constraints: for all g G , no two residents of g are invited to the same party; for all p P , the number of people invited to p is at most size ( p ) . (a) Formulate this problem as a linear-programming problem, much as we did for shortest paths. Any legal set of invitations should correspond to a feasible setting of the vari- ables for your LP, and any feasible integer setting of the variables in your LP should correspond to a legal set of invitations. What objective function maximizes the number of students invited? Solution: Let x p,g be a variable representing the number of invitations to party p P sent to residents of group g G . max p

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2 Handout 31: Problem Set 8 Solutions if x p,g = 1 will satisfy the requirements. (We do not permit the administration to send more than one invitation to a party p to the same student; thus we can send only one invitation to party p to group g .) Because the objective function p,g x p,g measures the number of invitations sent (and thus the number of students invited), an optimal setting of the variables for the LP therefore corresponds to a maximum number of invited students.
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## ps8sol - Introduction to Algorithms Massachusetts Institute...

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