09_Max_Flows_2 - 15.082 and 6.855J March 6, 2003 Maximum...

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1 15.082 and 6.855J March 6, 2003 Maximum Flows 2
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2 Network Reliability Communication Network What is the maximum number of arc disjoint paths from s to t? How can we determine this number? Theorem . Let G = (N,A) be a directed graph. Then the maximum number of arc-disjoint paths from s to t is equal to the minimum number of arcs upon whose deletion there is no directed s-t path. t s
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3 There are 3 arc-disjoint s-t paths s t 1 2 3 4 5 6 7 8 9 10 11 12
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4 Deleting 3 arcs disconnects s and t t 1 2 5 6 7 9 10 11 12 s 3 4 8 Let S = {s, 3, 4, 8}. The only arcs from S to T = N\S are the 3 deleted arcs.
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5 Node disjoint paths Two s-t paths P and P' f are said to be node- disjoint if the only nodes in common to P and P' are s and t). How can one determine the maximum number of node disjoint s-t paths? Answer: node splitting Theorem. Let G = (N,A) be a network with no arc from s to t. The maximum number of node- disjoint paths from s to t equals the minimum number of nodes whose removal from G disconnects all paths from nodes s to node t.
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6 There are 2 node disjoint s-t paths. s t 1 2 3 4 5 6 7 8 9 10 11 12
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7 Deleting 5 and 6 disconnects t from s? t 7 9 10 11 12 s 1 2 3 4 8 Let S = {s, 1, 2, 3, 4, 8} Let T = {7, 9, 10, 11, 12, t} There is no arc directed from S to T.
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8 Matchings An undirected network G = (N, A) is bipartite if N can be partitioned into N 1 and N 2 so that for every arc (i,j) either i N 1 and j N 2 . A matching in N is a set of arcs no two of which are incident to a common node. Matching Problem : Find a matching of maximum cardinality 1 2 3 4 5 6 7 8 9 10
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9 Transformation to a Max Flow Problem 1 2 3 4 5 6 7 8 9 10 s t Each arc (s, i) has a capacity of 1. Each arc (j, t) has a capacity of 1. Replace original arcs by pairs, and put infinite capacity on original arcs.
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10 Find a max flow 1 2 3 4 5 6 7 8 9 10 s t The maximum s-t flow is 4. The max matching has cardinality 4.
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11 Determine the minimum cut 2 5 7 9 10 t S = {s, 1, 3, 4, 6, 8}. T = {2, 5, 7, 9, 10, t}. There is no arc from {1, 3, 4} to {7, 9, 10} or from {6, 8} to {2, 5}. Any such arc would have an infinite capacity. 1 3 4 6 8 s
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12 Interpret the minimum cut 7 9 10 Look at the original nodes incident to the minimum cut. Every original arc is incident to one of them. 1 3 4 2 5 6 8 t
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13 Matching Duality 1 7 9 10 The maximum cardinality of a matching is the minimum number of nodes that “covers” all of the arcs.
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09_Max_Flows_2 - 15.082 and 6.855J March 6, 2003 Maximum...

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