# L07-mincut - Minimum Cost Flow and Minimum Cut Lecture 7...

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1 Minimum Cost Flow and Minimum Cut Lecture 7: Jan 31

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2 Flows    An  s-t   flow is a function  f   on the edges which satisfies: (capacity constraint)     (conservation of flows)    Value of the flow
3 Minimum Cost Flows Goal: Build a cheap network to satisfy the flow requirement.  A directed graph  G    A source vertex  s    A sink vertex    A capacity function  c  on the edges, i.e.  c:E->R  A cost function  w  on the edges, i.e.  w:E->R    A flow requirement  k Output: an  s-t   flow  f   of value k which  minimizes  f(e) w(e)  Σ Input:

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4 Special cases Shortest path : find a shortest path between  s   and  t     A minimum cost flow with  k = 1   Maximum flow : find a maximum flow between  s   and  t   Every edge in the original graph has cost  0 . Disjoint paths : connect  s   and  t   by  k    paths with min # of edges   Every edge in the original graph has cost  1  and capacity  1 .
5 Stuctures Recap Bipartite matchings Maximum flows  Shortest paths Minimum Cost Flows M-augmenting paths Residual graph augmenting paths ??? Weighted Bipartite matchings Negative cycle, Shorest augmenting path

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6 Residual Graph c(e) = 10 f(e) = 2 c(e) = 8 c(e) = 2 How about the  cost  of the flow?
7 Residual Graph c(e) = 10 f(e) = 2 c(e) = 8 c(e) = 2 c(e) = 10, w(e) f(e) = 2 c(e) = 8, w(e) c(e) = 2, -w(e) Max Flow Min-cost Flow

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8 Weighted Bipartite Matching
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## This note was uploaded on 10/13/2009 for the course CS 5150 taught by Professor Xulei during the Spring '09 term at University of Central Arkansas.

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L07-mincut - Minimum Cost Flow and Minimum Cut Lecture 7...

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