Algorithms, Lecture 3 on NP : Nondeterministic Polynomial Time · Defined NP-Complete: 1. X ∈NP,...

Algorithms,Lecture3onNP:NondeterministicPolynomialTime

ProblemXispolytimereducibletoYX≤PY

ifcansolveXusingpolycomputationandapolynumberofcallstoanalgorithmsolvingY.

“Uptopolyfactors,XisatleastaseasyasY”“Uptopolyfactors,YisharderthanX”

DefinedNP:decision(yes/no)problemscancheckcertificatesforyesanswersinptime

Havepolytimeproofcheckers:ApolytimealgorithmAsothatifX(s)=yes,

thereexistsawsothatA(s,w)=yes,andifX(s)=no,

forallw,A(s,w)=no

Mostproblemsofform:doesthereexistwsothatsatisfiesconditionsimpliedbys?

DefinedNP-Complete:1.X∈ NP,and2.XisNP-hard

XisNP-hardif:1.forallY∈ NP,Y≤P X,or2.CircuitSat ≤P X,or3.Y≤P X,forsomeNP-HardY,suchas

SAT,IndependentSet,VertexCover

Lastclass:

Lastclass:ProvedCircuitSAT isNP-Hard.

ProvedCircuitSAT ≤P SAT.

Infact,provedCircuitSAT ≤P 3-SAT

Where3-SATisSAT,buteachclausehasatmost3terms.

Lastclass:ProvedCircuitSAT isNP-Hard.

ProvedCircuitSAT ≤P SAT.

Infact,provedCircuitSAT ≤P 3-SAT

Where3-SATisSAT,buteachclausehasatmost3terms.Naturallyfollowsfromtheconstruction.

SAT≤P 3-SAT,aself-containedproof

AorBisequivalentto∃ys.t. (Aory)and(Bory)

So,givenaclausewithmorethan3terms,like

Introduceanewvariabley,andreplacewithclauses:

x1Vx2Vx3 Vx4

x1Vx2VyandyVx3 Vx4

SAT≤P 3-SAT,aself-containedproof

AorBisequivalentto∃ys.t. (Aory)and(Bory)

So,givenaclausewithmorethan3terms,like

Introduceanewvariabley,andreplacewithclauses:

x1Vx2Vx3 Vx4

x1Vx2VyandyVx3 Vx4

2-SATisinP

Alsopossibletoforceeachvariabletoappearatmost3times

Sayavariablexappearsinkclauses.

Replacexwithknewvariables,x1,…,xk,oneforeachclauseinwhichitappears.

Addclausesx1Vx2 ,x2Vx3 ,…,xk-1Vxk ,xk Vx1

Onlysatisfiedifallareequal.

Today

WillprovemoreproblemsareNP-complete:

3-coloringHamiltonianCycleTravellingSalespersonProblem

k-Coloring

GivenagraphG=(V,E),doesthereexistf:V→ {1,2,…,k}(colors)

Sothatforall(u,v)∈ Ef(u) f(v)?

3-colorable Not3-colorable

6=

3-colorable Not3-colorable

k-Coloring

GivenagraphG=(V,E),doesthereexistf:V→ {1,2,…,k}(colors)

Sothatforall(u,v)∈ Ef(u) f(v)?6=

k-ColoringisNP-Complete

ClearlyinNP,becausecancheckaproposedcoloringToproveNP-hard,willshow3-SAT≤P 3-Coloring

GivenacollectionofclausesC1,…,Ck,eachwithatmost3terms,onvariablesx1,…,xnproducegraphG=(V,E)thatis3-colorableiff theclausesaresatisfiable

3-ColoringisNP-Complete– variablegadgets

Create3specialnodes:T,F,B(base),andonenodeforeachterm:xi andxi

x1x1 x2x2 xnxn

Inevery3-coloring,oneofxi andxiiscoloredTandoneiscoloredF

T F

B

xnxnx1

Create3specialnodes:T,F,B(base),andonenodeforeachterm:xi andxi

x1 x2x2

Inevery3-coloring,oneofxi andxiiscoloredTandoneiscoloredF

3-ColoringisNP-Complete– variablegadgets

FT

B

Considerclausex1Vx2 Vxn

3-ColoringisNP-Complete– clausegadgets

x1 x2 xn

Claim:3-colorableiff termscoloredtosatisfyclause

T

B

3-ColoringisNP-Complete– clausegadgets

x1 x2 xn

Claim:3-colorableiff termscoloredtosatisfyclause

1.IftermsallcoloredF,thencannot3-color

T

B

xnx2x1

3-ColoringisNP-Complete– clausegadgetsClaim:3-colorableiff termscoloredtosatisfyclause

2.Ifsometermtrue,can3-color

T

B

x2x1 xn



T

B

xnx2x1



T

B

3-ColoringisNP-Complete

x1x1 x2x2 xnxnx3x3

x1Vx2 Vxn

3-colorableiffsatisfiable

FT

B

3-ColoringisNP-Complete

x1x1 x2x2 xnxnx3x3

x1Vx2 Vxn x2Vx3 Vxn

3-colorableiffsatisfiable

FT

B

HamiltonianCycle:Acycleinagraphthathitseachvertexonce.

DirectedHamiltonianCycle:same,butinadirectedgraph

DirectedHamCycleisNP-Complete

ClearlyinNP,becausecancheckifacycleisHamiltonian

ToproveNP-hard,willshowSAT≤P DirectedHamCycle

ProducedirectedgraphG=(V,E)thathasHamCycleiff theclausesaresatisfiable

Start:creategraphwith2n HamCycles,thencreategadgetstorestrictthem

1:

2:

3:

n:


s

t

1:

2:3:

n:

Mustgotop-to-bottom,andcantraverseeachrowleft-to-right(True)orright-to-left(False)


Mustgotop-to-bottom,andcantraverseeachrowleft-to-right(True)orright-to-left(False)

s

t

1:

2:3:

n:

Clausegadgets TrueFalse

s

t

1:

2:3:

n:

clausex1 forcestraversefirstrow

Require:nootheredgestouchverticesingadget

x1Vx2 Vxn


s

t

1:

2:3:

n:

clause

Forcestraverse1,or2,orn

x1Vx2 Vxn


s

t

1:

2:3:

n:

clause

Forcestraverse1,or2,ornRequire:nootheredgestouchverticesingadget

x1Vx2 Vxn


2:

clause

Toseemustcomebacktosamerow,notethatifdonotisnoHamiltonianpaththroughunuseddown-link

x1Vx2 Vxn


1:

2:

n:

clause

Forcestraverse1,or2,orn

x1Vx2 Vxn

Hamcycleiff satisfiable TrueFalse

s

t

1:

2:3:

n:

Pf.Ifsatisfiable,traverseinorderindicatedbyvars,pickingupeachclauseonceusingsometrueterm.

x1Vx2 Vx3

x1Vx2 Vxn

Hamcycleiff satisfiable TrueFalse

s

t

1:

2:3:

n:

x1Vx2 Vx3Pf.IfHamCycle,mustgotoptobot

assignvars bydirection

ifvisiteachclausenode,thenismadetruebytermonrowfromwhichmakethevisit.

voutvin

DirectedHamCycle≤PHamCycle

1. Indirectedproblem,answersameifreverseallarrows.

2.Totransformtoundirected,replaceeachvertexvwiththreevertices:vin,vbase,vout

vbuoutuin ub

u vReplacedirected(u,v)edgewith(uout ,vin )


uoutuin ub

Claim:Iftheseareonlyedgestoub,thenineveryHamiltoniancycleub mustbeadjacenttouin

Proof:ifitisnot,thenonceenterubcannotgetout

voutvin vbuoutuin ub

Replacedirected(u,v)edgewith(uout ,vin )

DirectedHamCycleinoriginal->HamCycle

woutwin wb

u v

w


voutvin vbuoutuin ub

Replacedirected(u,v)edgewith(uout ,vin )

Lemma:EveryHamCycleintheundirectedgraphmustgoin,base,out,in,base,out,in,base,out,etc,mustcorrespondtoaHamCyc indirectedgraph

woutwin wb

u v

w


TSP(TravellingSalespersonProblem)

Givennlocations,adistancefunctiond(u,v)andatotaldistanceD,doesthereexistatourthroughalllocationsoftotaldistanceatmostL?

http://www.tsp.gatech.edu/usa13509/usa13509_info.html

TSP(TravellingSalespersonProblem)

Givennlocations,adistancefunctiond(u,v)andatotaldistanceD,doesthereexistatourthroughalllocationsoftotaldistanceatmostL?

http://www.tsp.gatech.edu/usa13509/usa13509_sol.html

RL5915 optimal solutionAn optimal solution for RL5915 is given by the following tour, which has length 565530.

http://www.tsp.gatech.edu/rl5915/rl5915_sol.html

TSPisNP-complete

HamCycle≤PTSP

GivengraphG=(V,E),createonelocationforeachvertex,

d(u,v)=1if(u,v)∈ E2otherwise

Targetdistance=|V|

Atourofalllocationsthatreturnstostartandhastotallength|V|mustuseexactly|V|edgesofG

TSPisNP-complete

HamCycle≤PTSP



Thisisanabstractdistancefunction.Satisfiesd(u,v)=d(v,u)forallu,v andd(u,w)≤ d(u,v)+d(v,w)forallu,v,w

TSPisNP-complete

HamCycle≤PTSP



Thisisanabstractdistancefunction.

RemainsNP-hardforintegerpointsinplane.

IssuewithPlanarTSP

Ifinputislocationsofpoints,insteadofdistances

TheproblemisnotknowntobeinNP,becausedonotknowifcancomparedistancesinpolynomialtime.

Forintegersx1,…,xn integert,donothavepolytimealgorithmtotestif

X

i

pxi t

Algorithms, Lecture 3 on NP : Nondeterministic Polynomial Time · Defined NP-Complete: 1. X ∈NP,...

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