Introduction A polynomial instance

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Introduction A polynomial instance Reduction to the polynomial instance Conclusion Multicut is FPT Nicolas Bousquet Joint work with: Jean Daligault, Stephan Thomasse Nicolas Bousquet Multicut is FPT

attachment vertices

parameterized complexity

instance

binary tree

introduction parameterized complexity

fixed parameter

can branch

vertex

cover

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IntroductionApoylonimlanitsnaecduReioctotntpoheonyllaimtsniecnalusiConconasBousquNicolsituTPFuMtecitl

work

Joint

Thomass´e

with:JeanDaligault,Ste´phan

Multicut is FPT

Nicolas Bousquet

cutiulti

Introduction Parameterized complexity Multicut

sFPT

A polynomial instance

Reduction to the polynomial instance Vertex Multicut Reductions for one attachment vertex Two attachment vertices components

IpolyionAductntroeRecnatsnilaimonlypoheotntioctduulcnnoisaraPetemminoinalanstCoceluituctrizedcomplexityMuetMousqlasBNicoisnocnul4oC

ucedontithtoolepmonyilaiatsnCecnIntroductionApolnymoaiilsnatcnRetucitluMmatePnrasuoinolcxitympleedcoerizlosaNciuqteoBsultMuuticFPis

FPT A parameterized problem isFPT(Fixed Parameter Tractable) iﬀ there is an algorithm which runs in timePoly(n)∙f(k) for an instance of sizenand of parameterk.

oryaxy:xedgeckanCxvoreethtVeerniranbcaweceen.HerhcihwedicedothcnoorPiP:ftttocaneooesehhctex.nverrytrBinatpedfoeesomta:khncrakbt2coNis.henoiessleceetidtnheVertexCover.Deaercbkesenoyddnaetelheetgeeddjsa

Theorem Vertex Cover parameterized by the

Example

Parameterized complexity Multicut

FPT.

size of the solution

itucTPFstMuetiulsBlasqouolynthepontouctiRedeatcnilsnmoaiynolApontiucodtrnIusclniocnatnoCeaimosnil

RecnatsnilaimonyolApontiucodtrIncnCesnataiilnymoepoltothtioneducizercoedlemptyxilcnooisuraPntemauMtlcituitucitFsTP

Example

Proof : Pick an edgexy:xoryare in the Vertex Cover. Hence we can branch to decide which one is selected in the Vertex Cover. Decreasekby one and delete the edges adjacent to the choosen vertex.

Theorem Vertex Cover parameterized by the size of the solution is FPT.

arbk2tsoiN.sehcnousBlacoultMuesqanyriBofdetree:atmpthk

Example

Theorem Vertex Cover parameterized by the size of the solution is FPT.

Proof : Pick an edgexy:xoryare in the Vertex Cover. Hence we can branch to decide which one is selected in the Vertex Cover. Decreasekby one and delete the edges adjacent to the choosen vertex. Binary tree of depthk: at most 2kbranches.