MAXIMUM FLOW Max-Flow Min-Cut Theorem (Ford Fukerson’s Algorithm)
Maximum Flows and Minimum Cuts II - McGill School Of ...breed/2018COMP610/... · Max Flow Min Cut...
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Maximum Flows and Minimum Cuts II Lecture 6 COMP 610 January 24 2018
Transcript of Maximum Flows and Minimum Cuts II - McGill School Of ...breed/2018COMP610/... · Max Flow Min Cut...
MaximumFlowsandMinimumCutsII
Lecture6COMP610January242018
Recall:ACapacitatedNetworkDirectedGraphG=(V,E),nonnegativecapacityc(e)(sometimesu(e)isused)foreachedgeeuniquesourcesandsinkt,noedgesintosoroutoft,
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Recall:Ans-tFlow∀e∈Ehavef(e),0≤ f(e)≤c(e)(capacityconstraints)∀v∈V∑ 𝑓 𝑢𝑣 =)*∈+ ∑ 𝑓(𝑢𝑣)*)∈+ (flowconservationconstraints)
Recall:Ans-tCutisapartition(A,B)ofVwiths∈ 𝐴, 𝑏 ∈ 𝐵.Itscapacityc(A,B)=∑ 𝑐(𝑢𝑣))*∈+,)∈5,*∈6
Recall:Thevalueofaflowandflowacrossa cut
ForallS⊆ 𝑉: fout(S)=∑ 𝑓(𝑥𝑦))*∈+,)∈<,*∈=><
fin(S)=∑ 𝑓(𝑥𝑦))*∈+,)∈=><,*∈<
v(f)=fout({s})Foralls-tcuts(A,B),v(f)=fout(A)-fin(A)≤ c(A,B)Proof:v(f)=∑ (𝑓*∈5
out(v)-fin(v))=∑ 𝑓*∈5out(v)-∑ 𝑓?@*∈< (v)
=(fout(A)+∑ 𝑓(𝑣𝑤)*B∈+,*∈5,B∈5 )- (fin(A)+∑ 𝑓(𝑣𝑤)*B∈+,*∈5,B∈5 )=fout(A)-fin(A)≤ fout(A)≤ c(A,B)
Typeequationhere.
MaxFlowMinCutTheorem
Foreverycapacitatednetwork.Themaximumvolumeofans-tflowisequaltotheminimumvolumeofans-tcut.
Theorem7.14KTIfallthecapacitiesareintegerthenthereisamaximumvolumes-tflowfwhichisintegervalued. CanfindthisflowinO(v(f)|E|)time.
Recall:Matchings andCovers
AmatchingMisadisjointsetofedgesC⊆VisacoverifthereisnoedgewithbothendpointsinV-C.ForamatchingMandcoverCinanyG,|M|≤ |C|
ForBipartiteG:TheMaxSizeofaMatching=MinSizeofacover
MaximumNumberofEdge-DisjointPathsinaDirectedGraphG
V(G’)=V(G),E(G’)=E(G)-{xs|xs∈ 𝐸(𝐺)} − {𝑡𝑥|𝑡𝑥 ∈E(G)}Findainteger-valuedmaxvalues-tflowfinG’.Findasetofv(f)edgedisjointpathsfroms-tinG,onlyusingeiff(e)=1.Foranys-tcut(A,B),c(A,B)=|{ab∈ E|a∈ A,b∈ B}|.Foranys-tcut(A,B),removingtheset{ab∈ E|a∈ A,b∈ B}separatessfromt(i.e.destroysalls-tpaths).Theorem:Themaximumsizeofasetofedgedisjoints-tpathsinadirectedgraphequalstheminimumsizeofasetofedgeswhoseremovalseparatessfromt.
MaximumNumberofEdge-DisjointPathsinaDirectedGraphG
V(G’)=V(G),E(G’)=E(G)-{xs|xs∈ 𝐸(𝐺)} − {𝑡𝑥|𝑡𝑥 ∈E(G)}Findainteger-valuedmaxvalues-tflowfinG’.Findasetofv(f)edgedisjointpathsfroms-tinG,onlyusingeiff(e)=1.Foranys-tcut(A,B),c(A,B)=|{ab∈ E|a∈ A,b∈ B}|.Foranys-tcut(A,B),removingtheset{ab∈ E|a∈ A,b∈ B}separatessfromt(i.e.destroysalls-tpaths).Theorem:Themaximumsizeofasetofedgedisjoints-tpathsinadirectedgraphequalstheminimumsizeofasetofedgeswhoseremovalseparatessfromt.
MaximumNumberofEdge-DisjointPathsinaDirectedGraphG
V(G’)=V(G),E(G’)=E(G)-{xs|xs∈ 𝐸(𝐺)} − {𝑡𝑥|𝑡𝑥 ∈E(G)}Findainteger-valuedmaxvalues-tflowfinG’.Findasetofv(f)edgedisjointpathsfroms-tinG,onlyusingeiff(e)=1.Foranys-tcut(A,B),c(A,B)=|{ab∈ E|a∈ A,b∈ B}|.Foranys-tcut(A,B),removingtheset{ab∈ E|a∈ A,b∈ B}separatessfromt(i.e.destroysalls-tpaths).Theorem:Themaximumsizeofasetofedgedisjoints-tpathsinadirectedgraphequalstheminimumsizeofasetofedgeswhoseremovalseparatessfromt.
MaximumNumberofEdge-DisjointPathsinaDirectedGraphG
V(G’)=V(G),E(G’)=E(G)-{xs|xs∈ 𝐸(𝐺)} − {𝑡𝑥|𝑡𝑥 ∈E(G)}Findainteger-valuedmaxvalues-tflowfinG’.Findasetofv(f)edgedisjointpathsfroms-tinG,onlyusingeiff(e)=1.Foranys-tcut(A,B),c(A,B)=|{ab∈ E|a∈ A,b∈ B}|.Foranys-tcut(A,B),removingtheset{ab∈ E|a∈ A,b∈ B}separatessfromt(i.e.destroysalls-tpaths).Menger’s Theorem(1927):Themaximumsizeofasetofedgedisjoints-tpathsinadirectedgraphequalstheminimumsizeofasetofedgeswhoseremovalseparatessfromt.
MaximumNumberofEdge-DisjointPathsinanUndirectedGraphGV(G’)=V(G),E(G’)={uv,vu| uv ∈ 𝐸 𝐺 }−{xs|xs∈ 𝐸(𝐺)} − {𝑡𝑥|𝑡𝑥 ∈E(G)}AsetFofkedgedisjoints-tpathsinGyieldans-tflowofvaluekinG’.Asbefore,thereisanalgorithmwhichgivenaninteger-valuedflowfofvaluekinG’yieldsasetFofkdisjoints-tpathsinG’.EachsuchpathyieldsanundirectedpathinG.Thesearedisjointexceptthatthepathusinguv inG’mayintersectwiththepathusingvu.Howdowehandlethis????Iff(uv)=f(vu)=1,setf(uv)=f(vu)=0.ThenfindF.
MaximumNumberofEdge-DisjointPathsinanUndirectedGraphGV(G’)=V(G),E(G’)={uv,vu| uv ∈ 𝐸 𝐺 }−{xs|xs∈ 𝐸(𝐺)} − {𝑡𝑥|𝑡𝑥 ∈E(G)}AsetFofkedgedisjoints-tpathsinGyieldans-tflowofvaluekinG’.Asbefore,thereisanalgorithmwhichgivenaninteger-valuedflowfofvaluekinG’yieldsasetFofkdisjoints-tpathsinG’.EachsuchpathyieldsanundirectedpathinG.Thesearedisjointexceptthatthepathusinguv inG’mayintersectwiththepathusingvu.Howdowehandlethis????Iff(uv)=f(vu)=1,setf(uv)=f(vu)=0.ThenfindF.
MaximumNumberofEdge-DisjointPathsinanUndirectedGraphGV(G’)=V(G),E(G’)={uv,vu| uv ∈ 𝐸 𝐺 }−{xs|xs∈ 𝐸(𝐺)} − {𝑡𝑥|𝑡𝑥 ∈E(G)}AsetFofkedgedisjoints-tpathsinGyieldans-tflowofvaluekinG’.Asbefore,thereisanalgorithmwhichgivenaninteger-valuedflowfofvaluekinG’yieldsasetFofkdisjoints-tpathsinG’.EachsuchpathyieldsanundirectedpathinG.Thesearedisjointexceptthatthepathusinguv inG’mayintersectwiththepathusingvu.Howdowehandlethis????Iff(uv)=f(vu)=1,setf(uv)=f(vu)=0.ThenfindF.
MaximumNumberofEdge-DisjointPathsinanUndirectedGraphGV(G’)=V(G),E(G’)={uv,vu| uv ∈ 𝐸 𝐺 }−{xs|xs∈ 𝐸(𝐺)} − {𝑡𝑥|𝑡𝑥 ∈E(G)}AsetFofkedgedisjoints-tpathsinGyieldans-tflowofvaluekinG’.Asbefore,thereisanalgorithmwhichgivenaninteger-valuedflowfofvaluekinG’yieldsasetFofkdisjoints-tpathsinG’.EachsuchpathyieldsanundirectedpathinG.Thesearedisjointexceptthatthepathusinguv inG’mayintersectwiththepathusingvuinG’.Howdowehandlethis????Iff(uv)=f(vu)=1,setf(uv)=f(vu)=0.ThenfindF.
MaximumNumberofEdge-DisjointPathsinanUndirectedGraphGV(G’)=V(G),E(G’)={uv,vu| uv ∈ 𝐸 𝐺 }−{xs|xs∈ 𝐸(𝐺)} − {𝑡𝑥|𝑡𝑥 ∈E(G)}AsetFofkedgedisjoints-tpathsinGyieldans-tflowofvaluekinG’.Asbefore,thereisanalgorithmwhichgivenaninteger-valuedflowfofvaluekinG’yieldsasetFofkdisjoints-tpathsinG’.EachsuchpathyieldsanundirectedpathinG.Thesearedisjointexceptthatthepathusinguv inG’mayintersectwiththepathusingvuinG’.Howdowehandlethis????Iff(uv)=f(vu)=1,setf(uv)=f(vu)=0.ThenfindF.
MaximumNumberofEdge-DisjointPathsinanUndirectedGraphGV(G’)=V(G),E(G’)={uv,vu| uv ∈ 𝐸 𝐺 }−{xs|xs∈ 𝐸(𝐺)} − {𝑡𝑥|𝑡𝑥 ∈E(G)}AsetFofkedgedisjoints-tpathsinGyieldans-tflowofvaluekinG’.Asbefore,thereisanalgorithmwhichgivenaninteger-valuedflowfofvaluekinG’yieldsasetFofkdisjoints-tpathsinG’.EachsuchpathyieldsanundirectedpathinG.Thesearedisjointexceptthatthepathusinguv inG’mayintersectwiththepathusingvuinG’.Howdowehandlethis????Foralluv ∈ 𝐸 𝐺 , iff(uv)=f(vu)=1,setf(uv)=f(vu)=0.ThenfindF.
AirlineScheduling
Setofflights:(Origin,Destination,Departure Time,ArrivalTime)
Flightjisreachablefromflighti ifbetweenthearrivaltimeforflightiandthedeparturetimeforflightjthereistimetodueallscheduledmaintenanceandcleaning,andrelocatefromthedestinationofflightitotheoriginofflightj.
Wewanttofindoutifkplanescanhandlealltheflights.
AirlineScheduling
Setofflights:(Origin,Destination,Departure Time,ArrivalTime)
Flightjisreachablefromflighti ifbetweenthearrivaltimeforflightiandthedeparturetimeforflightjthereistimetodueallscheduledmaintenanceandcleaning,andrelocatefromthedestinationofflightitotheoriginofflightj.
Wewanttofindoutifkplanescanhandlealltheflights.
AirlineScheduling
Setofflights:(Origin,Destination,Departure Time,ArrivalTime)
Flightjisreachablefromflighti ifbetweenthearrivaltimeforflightiandthedeparturetimeforflightjthereistimetodueallscheduledmaintenanceandcleaning,andrelocatefromthedestinationofflightitotheoriginofflightj.
Wewanttofindoutifkplanescanhandlealltheflights.
AirlineScheduling
Setofflights:(Origin,Destination,Departure Time,ArrivalTime)
Flightjisreachablefromflighti ifbetweenthearrivaltimeforflightiandthedeparturetimeforflightjthereistimetodueallscheduledmaintenanceandcleaning,andrelocatefromthedestinationofflightitotheoriginofflightj.
Wewanttofindoutifkplanescanhandlealltheflights.
AirlineSchedulingViaEdgeDisjointPaths
Buildadigraphwithtwoverticesui andvi foreachflight.Thereisanedgefromui tovi foreveryi.Thereisanedgefromvi touj ifflightjisreachablefromflighti.Thereisasourcesandanedgefromstoeveryui.Thereisasinktandanedgefromeveryvi tot.ApathPfromstotpointsoutasetofflightswhichcanbehandledbyoneplane(thoseforwhichuivi isanedgeofP).Wecanscheduletheflightsusingkplanesifthereisasetofatmostkedge-disjointpathsofGusingtheedgeuivi foreveryflighti.
AirlineSchedulingViaEdgeDisjointPaths
Buildadigraphwithtwoverticesui andvi foreachflight.Thereisanedgefromui tovi foreveryi.Thereisanedgefromvi touj ifflightjisreachablefromflighti.Thereisasourcesandanedgefromstoeveryui.Thereisasinktandanedgefromeveryvi tot.ApathPfromstotpointsoutasetofflightswhichcanbehandledbyoneplane(thoseforwhichuivi isanedgeofP).Wecanscheduletheflightsusingkplanesifthereisasetofatmostkedge-disjointpathsofGusingtheedgeuivi foreveryflighti.
AirlineSchedulingViaEdgeDisjointPaths
Buildadigraphwithtwoverticesui andvi foreachflight.Thereisanedgefromui tovi foreveryi.Thereisanedgefromvi touj ifflightjisreachablefromflighti.Thereisasourcesandanedgefromstoeveryui.Thereisasinktandanedgefromeveryvi tot.ApathPfromstotpointsoutasetofflightswhichcanbehandledbyoneplane(thoseforwhichuivi isanedgeofP).Wecanscheduletheflightsusingkplanesifthereisasetofatmostkedge-disjointpathsofGusingtheedgeuivi foreveryflighti.
AirlineSchedulingViaEdgeDisjointPaths
Buildadigraphwithtwoverticesui andvi foreachflight.Thereisanedgefromui tovi foreveryi.Thereisanedgefromvi touj ifflightjisreachablefromflighti.Thereisasourcesandanedgefromstoeveryui.Thereisasinktandanedgefromeveryvi tot.ApathPfromstotpointsoutasetofflightswhichcanbehandledbyoneplane(thoseforwhichuivi isanedgeofP).Wecanscheduletheflightsusingkplanesifthereisasetofatmostkedge-disjointpathsofGusingtheedgeuivi foreveryflighti.
AirlineSchedulingViaEdgeDisjointPaths
Buildadigraphwithtwoverticesui andvi foreachflight.Thereisanedgefromui tovi foreveryi.Thereisanedgefromvi touj ifflightjisreachablefromflighti.Thereisasourcesandanedgefromstoeveryui.Thereisasinktandanedgefromeveryvi tot.ApathPfromstotpointsoutasetofflightswhichcanbehandledbyoneplane(thoseforwhichuivi isanedgeofP).Wecanscheduletheflightsusingkplanesifthereisasetofatmostkedge-disjointpathsofGusingtheedgeuivi foreveryflighti.
AirlineSchedulingViaEdgeDisjointPaths
Buildadigraphwithtwoverticesui andvi foreachflight.Thereisanedgefromui tovi foreveryi.Thereisanedgefromvi touj ifflightjisreachablefromflighti.Thereisasourcesandanedgefromstoeveryui.Thereisasinktandanedgefromeveryvi tot.ApathPfromstotpointsoutasetofflightswhichcanbehandledbyoneplane(thoseforwhichuivi isanedgeofP).Wecanscheduletheflightsusingkplanesifthereisasetofatmostkedge-disjointpathsofGusingtheedgeuivi foreveryflighti.
AirlineSchedulingViaBipartiteMatching
Foranyscheduleusingexactlykplanestohandlefflights,thecorrespondingkedgedisjointpathsusekedgesoutofs,kedgesintot,thefedgesoftheformuivi,andf-kedgesoftheformviuj whereflightjisreachablefromflighti.Theselastf-kedgesformamatching.Thedesiredkpathsexistpreciselyifthereisamatchingofsizef-kinthegraphG’whoseedgesetis{vjui|flighti isreachablefromflightj}.Sothenumberofplanesneededistheminimumkforwhichthereisamatchingofsizen-kinG’.Giventhematchingwecanfindthescheduleeasily,sothetimetofindanoptimalscheduleisO(f^3).
AirlineSchedulingViaBipartiteMatching
Foranyscheduleusingexactlykplanestohandlefflights,thecorrespondingkedgedisjointpathsusekedgesoutofs,kedgesintot,thefedgesoftheformuivi,andf-kedgesoftheformviuj whereflightjisreachablefromflighti.Theselastf-kedgesformamatching.Thedesiredkpathsexistpreciselyifthereisamatchingofsizef-kinthegraphG’whoseedgesetis{vjui|flighti isreachablefromflightj}.Sothenumberofplanesneededistheminimumkforwhichthereisamatchingofsizen-kinG’.Giventhematchingwecanfindthescheduleeasily,sothetimetofindanoptimalscheduleisO(f^3).
AirlineSchedulingViaBipartiteMatching
Foranyscheduleusingexactlykplanestohandlefflights,thecorrespondingkedgedisjointpathsusekedgesoutofs,kedgesintot,thefedgesoftheformuivi,andf-kedgesoftheformviuj whereflightjisreachablefromflighti.Theselastf-kedgesformamatching.Thedesiredkpathsexistpreciselyifthereisamatchingofsizef-kinthegraphG’whoseedgesetis{vjui|flighti isreachablefromflightj}.Sothenumberofplanesneededistheminimumkforwhichthereisamatchingofsizen-kinG’.Giventhematchingwecanfindthescheduleeasily,sothetimetofindanoptimalscheduleisO(f^3).
AirlineSchedulingViaBipartiteMatching
Foranyscheduleusingexactlykplanestohandlefflights,thecorrespondingkedgedisjointpathsusekedgesoutofs,kedgesintot,thefedgesoftheformuivi,andf-kedgesoftheformviuj whereflightjisreachablefromflighti.Theselastf-kedgesformamatching.Thedesiredkpathsexistpreciselyifthereisamatchingofsizef-kinthegraphG’whoseedgesetis{vjui|flighti isreachablefromflightj}.Sothenumberofplanesneededistheminimumkforwhichthereisamatchingofsizen-kinG’.Giventhematchingwecanfindthescheduleeasily,sothetimetofindanoptimalscheduleisO(f^3).
SequentialSchedulingViaBipartiteMatching
ImageSegmentationGraph.
GraphG=(V,E).Visthesetofpixels,Ejoinsneighbours.
ImageSegmentationProblem
• ForeachpixelI,have:ai-likelihoodi isintheforegroundandbi-likelihoodi isinthebackground.
• Foreachedgeij oftheImageSegmentationgraphhaveseparationpenaltypijincurredifwemakedifferentchoicesfori andj.• WewanttofindapartitionofthepixelsintoasetFofforegroundpixelsandasetBofbackgroundpixelssoastomaximize:
∑ 𝑎??∈Y +∑ 𝑏??∈6 −∑ 𝑝?[?[∈+,?∈Y,[∈6
• Thisisthesameasminimizing∑ 𝑎??∈6 +∑ 𝑏??∈Y +∑ 𝑝?[?[∈+,?∈Y,[∈6
ImageSegmentationProblem
• ForeachpixelI,have:ai-likelihoodi isintheforegroundandbi-likelihoodi isinthebackground.
• Foreachedgeij oftheImageSegmentationgraphhaveseparationpenaltypijincurredifwemakedifferentchoicesfori andj.• WewanttofindapartitionofthepixelsintoasetFofforegroundpixelsandasetBofbackgroundpixelssoastomaximize:
∑ 𝑎??∈Y +∑ 𝑏??∈6 −∑ 𝑝?[?[∈+,?∈Y,[∈6
• Thisisthesameasminimizing∑ 𝑎??∈6 +∑ 𝑏??∈Y +∑ 𝑝?[?[∈+,?∈Y,[∈6
ImageSegmentationProblem
• ForeachpixelI,have:ai-likelihoodi isintheforegroundandbi-likelihoodi isinthebackground.
• Foreachedgeij oftheImageSegmentationgraphhaveseparationpenaltypijincurredifwemakedifferentchoicesfori andj.• WewanttofindapartitionofthepixelsintoasetFofforegroundpixelsandasetBofbackgroundpixelssoastomaximize:
∑ 𝑎??∈Y +∑ 𝑏??∈6 −∑ 𝑝?[?[∈+,?∈Y,[∈6
• Thisisthesameasminimizing∑ 𝑎??∈6 +∑ 𝑏??∈Y +∑ 𝑝?[?[∈+,?∈Y,[∈6
ImageSegmentationProblem
• ForeachpixelI,have:ai-likelihoodi isintheforegroundandbi-likelihoodi isinthebackground.
• Foreachedgeij oftheImageSegmentationgraphhaveseparationpenaltypijincurredifwemakedifferentchoicesfori andj.• WewanttofindapartitionofthepixelsintoasetFofforegroundpixelsandasetBofbackgroundpixelssoastomaximize:
∑ 𝑎??∈Y +∑ 𝑏??∈6 −∑ 𝑝?[?[∈+,?∈Y,[∈6
• Thisisthesameasminimizing∑ 𝑎??∈6 +∑ 𝑏??∈Y +∑ 𝑝?[?[∈+,?∈Y,[∈6
ImageSegmentationViaMinCut
• Constructacapacited networkwhosevertexsetisthepixels,sandt.• Addanedgesi foreverypixeli withcapacityai• Addanedgeti foreverypixeli withcapacitybi• Foreverytwoneighbouring pixelsij,addedgesij andji bothwithcapacitypij• Thereisabijection betweens-tcutsandchoicesfortheforegroundFandbackground.Thecutcorrespondingto(F,B)is(s+F,t+B).• Wearelookingfortheminimumcapacitycut.
ImageSegmentationViaMinCut
• Constructacapacited networkwhosevertexsetisthepixels,sandt.• Addanedgesi foreverypixeli withcapacityai• Addanedgeti foreverypixeli withcapacitybi• Foreverytwoneighbouring pixelsij,addedgesij andji bothwithcapacitypij• Thereisabijection betweens-tcutsandchoicesfortheforegroundFandbackground.Thecutcorrespondingto(F,B)is(s+F,t+B).• Wearelookingfortheminimumcapacitycut.
ImageSegmentationViaMinCut
• Constructacapacited networkwhosevertexsetisthepixels,sandt.• Addanedgesi foreverypixeli withcapacityai• Addanedgeti foreverypixeli withcapacitybi• Foreverytwoneighbouring pixelsij,addedgesij andji bothwithcapacitypij• Thereisabijection betweens-tcutsandchoicesfortheforegroundFandbackground.Thecutcorrespondingto(F,B)is(s+F,t+B).• Wearelookingfortheminimumcapacitycut.
