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# DM OR - PowerPoint PPT Presentation

DM OR. Networks. Graphs : Koenigsberg bridges. Leonard Euler problem (1736). Nodes. Edges. Is there a walk which uses each edge exactly once ?. Build two more bridges . Each such walk has to enter and leave any given node . The degree of any node has to be even .

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## DM OR

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### DMOR

Networks

Graphs: Koenigsberg bridges

Leonard Euler problem (1736)

Euleriancycle

Hamiltonian cycle

Travelling salesman problem

TSP - cycle

Duration:

2 d 7 h 2 m Length:

3615 km

TSP - trail

Duration:

2 d 4 h 37 m

Length:

3436 km

TSP – a real walk

Duration:

29 d 11 h 21 m Length:

3485 km

Travelling salesman problem TSP
• Mathematically
• Variablexij= 1, ifthesalesmanpassesfromitoj, 0 otherwise
• Goingfrom one city to the same city isforbidden
• Isthisall ???
TSP
• Isthisall?
• Let’ssayoursolutionwith 5 citiesisx12=x23=x31=x45=x54=1
• Itsatisfiesalltheconstraints. But itinvolvessubtours (cyclesinvolvingfewerthanallcities)
Subtourelimination
• For twocities
• For threecities
• For fourcities
• Etc.
• In practicalimplementationtoo many constraints: with30 citiestherewould be 870 constraintseliminatingonlysubtours of length 2
Subtoureleimination– secondapproach
• Introducenonnegativevariablesui:
• Subtoursareeliminated
• How many suchconstraints?
• (N-1)2-N, i.e. with30 citiestherewould be 812constraints.
Game
• http://www.tsp.gatech.edu/games/index.html
Introduction to networks
• Twomainelements:
• arcs/edges
• nodes
• A graphis a structureconsisting of nodes and arcsbewteennodes
• A directedgraph(a digraph)is a graphinwhicharcshave a givendirection
• A networkis a graph(ordigraph), inwhicharcshave a flowassigned to it
• Simple examples:
Introduction to networks
• Chainis a sequence of arcsconnectingtwonodesiandj, e.g.figure on theright: ABCE, ADCE
• Pathis a sequence of directedarcsconnectingtwonodes , e.g. figure on therightABDE, but not ABCE
• Cycle is a chain, whichconnects a nodewiththe same nodewithoutanyrepetition (retracing) e.g. figure on therightABCEDA, but not ABCDECBA
• Connectedgraph/network hasonly one part

Graph

Directedgraph

Introduction to networks
• Tree – a connectedgraphwhichdoes not havecycles.
• Spanningtreeis a treechosenamongarcsinthegraph so thatallnodesinthetreewereconnected

Twospanningtrees

Twotrees

• Flowcapacity – upper (sometimesalsolower) limit for theflowat a givenarcinthenetwork, e.g. maximalnumber of cars thatcan pass per minuteat a givenroad
• Sourceis a nodewhichintroduces a flowintothenetwork
• Sinkis a nodewhichtakestheflow out of thenetwork
The shortest route problem
• Formulation:For a givengraphinwhicheveryarcisassignedwith a distancebewteenthetwonodesitconnects, whatistheshortestpathbetweennode i and j.
• Example: Whatistheshortestroutebewteen A and H?

Enumeration – impractical

Dijkstraalgorithm

Dijkstraalgorithm

http://optlab-server.sce.carleton.ca/POAnimations2007/DijkstrasAlgo.html

Minimum spanning tree
• Formulation:For a givengraph, inwhicheveryarcisassignedwith a distancebetweenthetwonodesitconnects, find a spanningtreethathasminimaltotallength.
• Example: Findminimallength for a wirethatconnectsalltheofficesinthebuildingwhenalltheavailablewirepathsaregiven.
• Algorithm:

http://optlab-server.sce.carleton.ca/POAnimations2007/MinSpanTree.html

An example of the so calledgreedy algorithm –itdoeswhat’sbestin a given step not lookingattheotherstages of the problem (usuallyineffective – hereitiseffective!)

One can do a maximalspanningtreethe same way

Maximum flow and the minimum cut
• Formulation:Whatisthemaximalflowbetweentwogivennodesin a graph? Eachnodeisassignedwith a maximalflow.
• Example: Find a maximalflow of cars from an underground parking lot downtown to themotorwayentrance.
• Eacharcisassignedwith a maximalsimultaneousflowbetweenthetwonodesitconnects
• Maximalflowmaydifferdepending on theflowdirection (e.g. one-waystreets)

Examplesolution:

4 cars/m on route A-D-E-G

3 cars/m ojnroute A-B-E-G

4 cars/m on route A-C-F-G

Total flowbetween A and G is11 cars/m

Isthisoptimal???

Maximum flow and the minimum cut
• Algorithm: Ford and Fulkerson (Canadian Journal of Mathematics 1956)

http://optlab-server.sce.carleton.ca/POAnimations2007/MaxFlow.html

Maximumflow/minimum cut
• In Ford Fulkersonalgorithm, why do we need to addflowintheoppositedirection?
• Accountingconventionthatkeepstrack of theflowthat, ifnecessary, can be reversed.
Maximumflow/minimum cut
• Maximumflowiscloselyrelated to minimum cut:
• A cutin a graphis a set of directedarcswhichcontainsatleast one arcineverypossiblepathfromthesource to thesink. If we removearcsfrom a givencut, theflowfromthesource to thesink will no longer be possible.
• Cutvalueisthe sum of allflowcapacities (directionfromthesource to thesink) for eacharcin a cut.
• Possiblecutswithcutvaluesindicated on them