16-max-flow-handout.pdf - Algorithm Design Techniques...

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CMPSCI 311: Introduction to Algorithms Lecture 16: Network Flows Akshay Krishnamurthy University of Massachusetts Last Compiled: April 3, 2018 Algorithm Design Techniques Greedy Divide and Conquer Dynamic Programming Network Flows Network Flow Previous topics (greedy, dynamic programming, divide and conquer etc.) were design techniques. Network flow relates to a specific class of problems with many applications Direct applications : commodities in networks transporting food on the rail network packets on the internet gas through pipes Indirect applications : Matching in graphs Airline scheduling Baseball elimination Plan : design and analyze algorithms for max-flow problem , then apply to solve other problems First, a Story About Flow and Cuts Key theme : flows in a network are intimately related to cuts Soviet rail network in 1955 On the history of the transportation and maximum flow problems. Alexander Schrijver, Math Programming, 2002. Capacity v 1 v 2 v 3 v 4 t s 16 13 14 9 12 4 7 20 4 Capacity/Flow v 1 v 2 v 3 v 4 t s 16/11 13/8 14/11 9/4 12/12 4/1 7/7 20/15 4/4
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Defining Flows Flow network Directed graph Source node s and target node t Edge capacities c ( e ) 0 Flow Capacity Constraints: 0 f ( e ) c ( e ) on each edge Conservation Constraints: f in ( s ) = 0 , f out ( t ) = 0 , v V \{ s, t } f in ( v ) = f out ( v ) where f in ( v ) = e in to v f ( e ) and f out ( v ) = e out of v f ( e ) Max flow problem: find a flow of maximum value v ( f ) = f out ( s ) Designing a Max-Flow Algorithm Something that doesn’t work: Repeatedly choose paths and
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  • Fall '09
  • Flow network, Maximum flow problem, Max-flow min-cut theorem, New Flow, Old Flow, Flow f

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