# lec13 - 15.053 z April 3 2007 The maximum flow problem z...

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1 15.053 April 3, 2007 z The maximum flow problem z See class notes on website.

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Quotes of the day You get the maxx for the minimum at TJ Maxx. -- ad for a clothing store This was the most unkindest cut of all -- Shakespeare in Julius Caesar Act 3 2
Goal of this lecture z Introduce the maximum flow problem z Discuss applications of flows z Present the Ford-Fulkerson max flow algorithm z Present the max-flow min-cut result 3

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4 The Maximum Flow Problem z Directed Graph G = (N, A). Source s Sink t Capacities u ij on arc (i,j) Maximize the flow out of s, subject to Flow out of i = Flow into i, for i s or t. A Network with Arc Capacities s 1 t 2 4 1 2 3 1 1 2 3 4 The max flow problem is to send as much flow from the source node s to the sink node t as possible subject to the following constraints: 1. The flow into node i is the flow out of node i for nodes i s, t. 2. The flow x ij on arc (i, j) satisfies 0 x ij u ij , where u ij is the capacity of arc (i, j).
5 Feasible flows z Flow in arc (i, j) is at most its capacity z Total flow into node 1 = total flow out of node z Total flow into node 2 = total flow out of node 2 A feasible flow s 1 t 2 1 1 1 1 0 1 2 3 4 capacities For most of the lecture, we will represent capacities using the color scheme given here and given line thicknesses. We do this so that our networks are not so “busy” with so many numbers. It is confusing to have to look at both capacities and flows at the same time.

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6 Two Infeasible flows s 1 t 2 1 2 1 2 0 1 2 3 4 capacities The flow out of node 1 exceeds the flow in. s 1 t 2 1 0 1 1 1 The capacity of arc (s, 2) is violated. This slide gives the two ways that a flow can be infeasible other than the flow being negative.
7 A feasible flow s 1 t 2 2 0 1 1 1 1 2 3 4 capacities A maximum flow s 1 t 2 3 1 2 2 1

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On representing capacities z Usually capacities are just listed as numbers. z We are using colors and thicknesses in this lecture to make the animations easier to follow. z On any homework set, just use numbers. Don’t use colors or thicknesses. 8
An application of maximum flow z In a college dormitory, students want to download a movie. They want to know how many minutes of the movie they can download each second. s 1 t 2 This is a straightforward application of max flow in that we want to send as many packets per second from the download site to the dorm. 9

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s 1 t 2 10 6 8 10 1 10 An application of maximum flow Starbucks has decided to create a slurry pipeline to send coffee from Seattle to retailers around the country. How much coffee can it ship to MIT every day? 1 2 4 1 3 2 1 s MIT This is a silly application. It is motivated by an April fools joke by NPR radio about 10 years ago. They had a story that sounded fully serious about a pipeline for shipping hot coffee from Seattle to the East Coast. They went into great detail about the legal rights concerning putting pipes on people’s properties and about the expenses involved.
The maximum matching problem M I CT D CY G P M Can you match the stars of Grey’s

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lec13 - 15.053 z April 3 2007 The maximum flow problem z...

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