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CMPSCI 611: Advanced Algorithms Micah Adler

Summary: CMPSCI 611: Advanced Algorithms
Micah Adler
Problem Set 3 Out: October 31, 2001
Due: November 7, 2001
1. Consider a hockey league with n teams T 1 : : : Tn . At a certain point in the season, we want to determine
if team T 1 has been mathematically eliminated. In other words, if, no matter what happens in the
remaining games, some other team will end up with more wins than team T 1 . In this league, there are
no ties.
The input will consist of a set of integers w 1 : : : wn , where w i is the number of wins that team T i has
thus far in the season, as well as c ij , for 1  i < j  n, which represents the number of remaining
games between teams i and j.
(a) Show that the problem of deciding whether team T 1 is mathematically eliminated can be formulated
as a maximum ow problem.
Hint: rst describe why you can assume that T 1 wins the rest of its games. Then focus on the values
` 2 : : : ` n , where ` i is the number of losses that team T i must have to not eliminate team T 1 .
(b) By applying the max- ow min-cut theorem to your network from part (a) provide and prove
correct a simple necessary and suĂcient condition for team T 1 being mathematically eliminated. This
condition should involve the existence of a subset of teams with a certain property.
2. [CLR] Problem 27-2 (page 626 of rst edition) Minimum path cover.
3. Given an undirected graph G = (V; E), and vertices s and t, let's say we want to nd a minimum


Source: Adler, Micah - Department of Computer Science, University of Massachusetts at Amherst


Collections: Computer Technologies and Information Sciences