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# Slides7_8 - Lecture 7/8 Maximum Flow Problems The Maximum...

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1 Lecture 7/8 Maximum Flow Problems The Maximum Flow problem Given a directed network G=(V,E) with two special nodes s ( source ) and t ( sink ). We want to ship commodities from s to t. Each arc has a capacity c ij 0 associated to it. Problem: Find the maximum amount of commodities that can be shipped through the network from s to t. s 1 2 t 1 1 1 4 4 1 1 Example:

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2 Some applications Railway/Road network: How many tons of commodities can be shipped from s to t? Airline scheduling: Flights from s to t with stops (Winston, Example 4, page 421). How many flights can be scheduled? Internet traffic: How many data packages can be sent over a network from s to t? Matchmaking: Matching the maximum number of compatible male and female dance partners (Winston, Example 5, page 422). Baseball Elimination: Which team has (still) a chance to finish the season with most wins? (Discussed later) Etc. 50’s: Soviet railway system A. Schrijver: On the history of the transportation and maximum flow problems http://homepages.cwi.nl/~lex/files/histtrpclean.pdf History of MaxFlow problems
3 We will discuss the Ford-Fulkerson Method (Winston), also known as Ford-Fulkerson labeling algorithm. Ford-Fulkerson Method General idea: “Push” as many commodities as possible from s to t. Try to increment iteratively the flow by using “suitable” s-t-paths.

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4 Ford-Fulkerson Method: Example s 1 2 t 1 1 1 4 4 1 Flow value =0 Step 0: s 1 2 t 1, 1 1 1 , 1 4 4 1, 1 Flow value =1 Step 1: s 1 2 t 1, 1 1, 1 1 , 1 1, 4 1, 4 1, 1 Flow value =2 Step (c): s 1 2 t 1, 1 1 1 , 1 4 4 1, 1 Flow value =1 Step (a): ITERATION 1 s 1 2 t 1, 1 1 1 , 1 4 4 1, 1 Labeling
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Slides7_8 - Lecture 7/8 Maximum Flow Problems The Maximum...

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