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Max-Flow Min-Cut Applications: Playoff Elimination.

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Presentation on theme: "Max-Flow Min-Cut Applications: Playoff Elimination."— Presentation transcript:

1 Max-Flow Min-Cut Applications: Playoff Elimination

2 Quick Review of MLB Playoffs Two leagues, American and National: independent Three divisions per league Who makes playoffs? –Top team in each division –Team with best record among all second- place teams in league (“wildcard”)

3 End-of-Season Scenario American League East WonLostPct.GB New York77610.558- Baltimore73650.5294 Boston71680.5116.5 Toronto63760.45314.5 Detroit50890.36027.5 Central WonLostPct.GB Cleveland81570.587- Chicago75650.5367 Minnesota69700.49612.5 Milwaukee68720.48614 Kansas City65750.46417 West WonLostPct.GB Texas78600.565- Seattle72660.5226 Oakland67740.47512.5 California64750.46014.5  New York Wins East  Cleveland Wins Central  Texas Wins West  Chicago Wins Wildcard

4 Can Detroit Win This Division? WLGBGR New York7559-28 Baltimore7163428 Boston69666.527 Toronto637212.527 Detroit498626.527 8/30/96

5 Remaining Series in the AL East TeamsGames Baltimore vs. Boston2 Baltimorevs. New York3 Baltimore vs. Toronto7 Boston vs.New York8 New York vs. Toronto7 8/30/96

6 Proof of Detroit’s Elimination Try to create a scenario where Detroit wins the division. Detroit can finish with at most 76 wins. New York already has 75 wins. If New York wins 2 or more games, Detroit is eliminated. TeamWL New York7785 Detroit7686

7 Scenarios Where Detroit Can Win the Division. Case 1: New York loses all it’s remaining games. Case 2: New York wins 1 more game.

8 Case 1: New York Loses All It’s Remaining Games. New York plays Boston 8 more times. Boston already has 69 wins. Implies Boston finishes ahead of Detroit. Detroit is eliminated in Case 1. TeamWL Boston7785 Detroit7686 New York7587

9 Case 2: New York Wins Only 1 More Game New York must win 1 game against Boston. 75 + 1= 76. Boston cannot beat any teams other than New York. 69 + (8-1) = 76. Implies that Detroit, Boston and New York each finish with 76 wins. TeamWL Boston7686 Detroit7686 New York7686 What about Baltimore ?

10 Baltimore Already has 71 wins. Plays New York 3 more times. Plays Boston 2 more times. Baltimore will finish with at least 76 wins. A 4-way tie for first is possible, but only if Baltimore only beats New York and Boston. What about Toronto ? TeamWL Baltimore7686 Boston7686 Detroit7686 New York7686

11 Toronto Already has 63 wins. Plays New York 7 more times. Plays Baltimore 7 more times. Previous assumptions imply Toronto finishes with at least 77 wins. Detroit cannot win the division in Case 2 either. TeamWL Toronto7785 Baltimore7686 Boston7686 Detroit7686 New York7686

12 Literature Review B.Schwartz, 1966 A. Hoffman and J. Rivlin, 1967 –Solve First Place Elimination Problem via Network Flows L. Robinson, 1991 –Extensions to Other Sports D. Gusfield and C. Martel, 1992 –Minimum Number of Games to Avoid Elimination From First Place K. Wayne, 2001 –First Place Elimination Number via network flows

13 Detroit’s Elimination Network Team Nodes Game Nodes Bal vs. Bos. N.Y. vs. Tor. Bal. vs. N.Y. s 7 2 8 Bal. vs. Tor. 3 Bos. vs. N.Y. 7 Bal Bos N.Y. Tor Arcs with  capacity. t 76-71=5 76-69=7 76-75=1 76-63=13

14 Detroit’s Elimination Network Team Nodes Game Nodes Bal vs. Bos. N.Y. vs. Tor. Bal. vs. N.Y. s 7 2 8 Bal. vs. Tor. 3 Bos. vs. N.Y. 7 Bal Bos N.Y. Tor u[S,T] = 26 < # wins remaining = 27 t 5 7 1 13

15 More Interesting Questions! I. Adler, A. Erera, D. Hochbaum and E. Olinick, 2002 – “Clinching” Questions –Is my team guaranteed to win division? Make playoffs? “Magic Number” Questions –How many games must my team to win to clinch? –How many games must they win to avoid elimination?

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