GivenagraphG=(V,E)andasetS⊆V,asetU⊆

GivenagraphG=(V,E)andasetS⊆V,asetU⊆Visasubsetfeedbackvertexsetof(G,S)ifnocycleinG[VU]containsavertexofS.AsubsetfeedbackvertexsetUisminimalifnosubsetfeedbackvertexsetof(G,S)isapropersubsetofU.TheSubsetFeedbackVertexSetproblemtakesasinputG,S,andanintegerk,andthequestioniswhether(G,S)hasasubsetfeedbackvertexsetofcardinalityatmostk.Intheweightedversionoftheproblem,everyvertexofGhasaweight,andthequestioniswhetherthereisasubsetfeedbackvertexsetoftotalweightatmostk.
SubsetFeedbackVertexSetwasintroducedbyEvenetal.[5],anditgeneralizesseveralwell-studiedproblems.WhenS=V,itisequivalenttotheclassicalFeedbackVertexSetproblem[12],andwhen|S|=1,itgeneralizestheMultiwayCutproblem[8].WeightedSubsetFeedbackVertexSetadmitsapolynomial-timeconstant-factorapproximationalgorithm[5].Theunweightedversionoftheproblemisfixedparametertractable[3].TheonlyexactalgorithmknownforitsweightedversionisbyFominetal.[8]anditrunsinO(1.8638n)timeandsolvestheproblembyenumeratingallminimalsubsetfeedbackvertexsets.
LetusbrieflycompareSubsetFeedbackVertexSettoitsmorewidelyknownrestrictionFeedbackVertexSet.Theun-weightedversionofFeedbackVertexSetcanbesolvedintimeO(1.7347n)[10],whereasthebestknownalgorithmforitsweightedversionrunsintimeO(1.8638n)andenumeratesallminimalfeedbackvertexsets[6].FeedbackVertexSet

ผลลัพธ์ (ไทย) 1:

GivenagraphG = (V, E) andasetS⊆V, asetU⊆Visasubsetfeedbackvertexsetof (G, S) [VU] ifnocycleinG containsavertexofS.AsubsetfeedbackvertexsetUisminimalifnosubsetfeedbackvertexsetof (G, S) isapropersubsetofU.TheSubsetFeedbackVertexSetproblemtakesasinputG,S,andanintegerk,andthequestioniswhether (G, S) hasasubsetfeedbackvertexsetofcardinalityatmostkIntheweightedversionoftheproblem, everyvertexofGhasaweightandthequestioniswhetherthereisasubsetfeedbackvertexsetoftotalweightatmostk.
SubsetFeedbackVertexSetwasintroducedbyEvenetal[5], anditgeneralizesseveralwell-studiedproblemsWhenS = V, itisequivalenttotheclassicalFeedbackVertexSetproblem [12], andwhen|S| = 1, itgeneralizestheMultiwayCutproblem [8]WeightedSubsetFeedbackVertexSetadmitsapolynomial-timeconstant-factorapproximationalgorithm [5]Theunweightedversionoftheproblemisfixedparametertractable [3]TheonlyexactalgorithmknownforitsweightedversionisbyFominetal[8]anditrunsinO(1.8638n)timeandsolvestheproblembyenumeratingallminimalsubsetfeedbackvertexsets.
LetusbrieflycompareSubsetFeedbackVertexSettoitsmorewidelyknownrestrictionFeedbackVertexSet.Theun-weightedversionofFeedbackVertexSetcanbesolvedintimeO(1.7347n)[10],whereasthebestknownalgorithmforitsweightedversionrunsintimeO(1.8638n) andenumeratesallminimalfeedbackvertexsets [6]FeedbackVertexSet

ผลลัพธ์ (ไทย) 3:

givenagraphg = ( V , E ) andasets ⊆ V , asetu ⊆ visasubsetfeedbackvertexsetof ( G , S ) ifnocycleing [ V / u ] containsavertexofs . asubsetfeedbackvertexsetuisminimalifnosubsetfeedbackvertexsetof ( G , S ) isapropersubsetofu . andanintegerk thesubsetfeedbackvertexsetproblemtakesasinputg , s , , andthequestioniswhether ( G , S ) hasasubsetfeedbackvertexsetofcardinalityatmostk . intheweightedversionoftheproblem everyvertexofghasaweight , ,andthequestioniswhetherthereisasubsetfeedbackvertexsetoftotalweightatmostk .
subsetfeedbackvertexsetwasintroducedbyevenetal [ 5 ] , anditgeneralizesseveralwell studiedproblems . เมื่อไหร่ = v itisequivalenttotheclassicalfeedbackvertexsetproblem [ 12 ] , หรือ | S | = 1 , itgeneralizesthemultiwaycutproblem [ 8 ] . เวลาคงที่ factorapproximationalgorithm weightedsubsetfeedbackvertexsetadmitsapolynomial [ 5 ]theunweightedversionoftheproblemisfixedparametertractable [ 3 ] theonlyexactalgorithmknownforitsweightedversionisbyfominetal [ 8 ] anditrunsino ( 1.8638n ) timeandsolvestheproblembyenumeratingallminimalsubsetfeedbackvertexsets .
letusbrieflycomparesubsetfeedbackvertexsettoitsmorewidelyknownrestrictionfeedbackvertexset . โรงไฟฟ้า weightedversionoffeedbackvertexsetcanbesolvedintimeo ( 1.7347n ) [ 10 ]whereasthebestknownalgorithmforitsweightedversionrunsintimeo ( 1.8638n ) andenumeratesallminimalfeedbackvertexsets feedbackvertexset [ 6 ]