Lecture9

# Lecture9 - CME 305 Discrete Mathematics and Algorithms...

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Unformatted text preview: CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi ([email protected]) February 2, 2010 Lecture 9: Applications of Network Flow Problem In this lecture, we discuss three problems that can be modeled as network flow problems; (i) baseball elimination, (ii) project scheduling, (iii) airline flight scheduling. Chapter 7 of the textbook (Kleiberg and Tardos [KT]) discusses many other applications of network flow to solve different types of problems. 1 Baseball Elimination Problem: Suppose we are in the middle of a baseball season where each team T i , 1 ≤ i ≤ n has won W i games so far and thus has W i points (recall that in baseball each game has one point and we cannot have a tie.). Let G 1 ,G 2 ,...,G k be the schedule of the remaining games, where each G i is an unordered pair of teams. Question: Given T i , W i , 1 ≤ i ≤ n , and G 1 ,G 2 ,...,G k , can we predict that T 1 does or not does not have a chance to have the top score at the end of the season? if T 1 has a chance of being champion, how can we find a sequence of outcomes (i.e. results of G 1 ,G 2 ,...,G k ) such that T 1 reaches the top rank at the end of the season? Answer: Let g ( i ) be the number of games that team i will play until the end of the season. Let W * 1 = W 1 + g (1) which is the maximum final score of T 1 . If there exists....
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Lecture9 - CME 305 Discrete Mathematics and Algorithms...

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