Asymmetryin k -CenterVariants
IngeLiGørtz1 andAnthonyWirth2
1 TheoryDepartment,TheITUniversityofCopenhagen,Denmark. inge@it-c.dk
2 DepartmentofComputerScience,PrincetonUniversity. awirth@cs.princeton.edu
Abstract. Thispaperexploresthreeconcepts:the k -centerproblem, someofitsvariants,andasymmetry.The k -centerproblemisafundamentalclusteringproblem,similartothe k -medianproblem.Variantsof k -centermaymoreaccuratelymodelreal-lifeproblemsthantheoriginal formulation.Asymmetryisasignificantimpedimenttoapproximationin manygraphproblems,suchas k -center,facilitylocation, k -medianand theTSP.
Wedemonstratean O (log ∗ n)-approximationalgorithmfortheasymmetric weighted k -centerproblem.Here,theverticeshaveweightsand wearegivenatotalbudgetforopeningcenters.Inthe p-neighbor varianteachvertexmusthave p (unweighted)centersnearby:wegivean O (log ∗ k )-bicriteriaalgorithmusing2k centers,forsmall p. Finally,thefollowingthreeversionsoftheasymmetric k -centerproblem weshowtobeinapproximable: priority k -center, k -supplier,and outliers withforbiddencenters
1Introduction
Imagineyouhaveadeliveryservice.Youwanttoplaceyourdeliveryhubsat locationsthatminimizethemaximumdistancebetweencustomersandtheir nearesthubs.Thisisthe k -centerproblem —atypeofclusteringproblemthatis similartothefacilitylocationand k -medianproblems.Themotivationforthe asymmetric k -centerproblem,inourexample,isthattrafficpatternsorone-way streetsmightcausethetraveltimefromonepointtoanothertodifferdepending onthedirectionoftravel.Traditionally,the k -centerproblemwassolvedinthe contextofametric;inthispaperweretainthetriangleinequality,butabandon thesymmetry.
Symmetryisavitalconceptingraphapproximationalgorithms.Veryrecently,the k -centerproblemwasshowntobe Ω (log ∗ n)hardtoapproximate[6, 7],eventhoughthesymmetricversionhasafactor2approximation.Moreover,facilitylocationand k -medianbothhaveconstantfactoralgorithmsinthe
PartofthisworkwasperformedwhilevisitingPrincetonUniversity. SupportedbyaGordonWuFellowship,aDIMACSSummerResearchFellowship, andNSFITRgrantCCR-0205594.
S.Aroraetal.(Eds.):APPROX2003+RANDOM2003,LNCS2764,pp.59–70,2003. c Springer-VerlagBerlinHeidelberg2003
60IngeLiGørtzandAnthonyWirth
symmetriccase,butareprovably Ω (log n)hardtoapproximatewithoutsymmetry[1].Thetravelingsalesmanproblemisalittlebetter,inthatno Ω (log n) hardnessisknown,butwithoutsymmetrynoalgorithmbetterthan O (log n) hasbeenfoundeither.
Definition1(k -Center). Given G =(V,E ),acompletegraphwithnonnegative(butpossiblyinfinite)edgecostsandapositiveinteger k ,findaset S of k vertices,called centers,withminimumcoveringradius.Thecoveringradiusofa set S istheminimumdistance R suchthateveryvertexin V iswithindistance R ofsomevertexin S
KarivandHakimi[11]showedthatthe k -centerproblemis NP -hard.Withoutthetriangleinequalitytheproblemis NP -hardtoapproximate;wehenceforthassumethattheedgecostssatisfythetriangleinequality.
Theasymmetric k -centerproblemhasproventobemuchmoredifficultto understandthanitssymmetriccounterpart.HsuandNemhauser[10]showed thatthe k -centerproblemcannotbeapproximatedwithinafactorof(2 ) unless P = NP .In1985HochbaumandShmoys[8]provideda(bestpossible) factor2algorithmforthesymmetric k -centerproblem.In1996Panigrahyand Vishwanathan[16,13]gavethefirstapproximationalgorithmfortheasymmetric problem,withfactor O (log ∗ n).Archer[2]proposedtwo O (log ∗ k )algorithms basedonmanyoftheideasin[13].Wenowknow[6,7]thatthesealgorithms areasympotitcallythebestpossible.
Variantsofthe k -CenterProblem Anumberofvariantsofthe k -center problemhavebeenexploredinthecontextofsymmetricgraphs.Perhapssome deliveryhubsaremoreexpensivetoestablishthanothers: insteadofarestriction onthenumberofcenterswecanuse,eachvertexhasaweightandwehavea budget W ,thatlimitsthetotalweightofcenters.HochbaumandShmoys[9]producedafactor3algorithmforthis weighted k -centerproblem.Thishasrecently beenshowntobetight[6].
HochbaumandShmoys[9]alsostudiedthe k -supplierproblem wherethe vertexsetissegregatedintosuppliers andcustomers.Onlysuppliervertices canbecentersandonlythecustomerverticesneedtobecovered.Hochbaum andShmoysgavea3-approximationalgorithmandshowedthatthisisthebest possible.
Khulleretal.[12]investigatedthe p-neighbor k -centerproblem whereeach vertexmusthave p centersnearby.Thisproblemis motivatedbyneedtoaccount forfacilityfailures:evenifupto p 1facilitiesfail,everydemandpointhasa functioningfacilitynearby.Theygavea3-approximationalgorithmforall p, andabestpossible2-approximationalgorithmwhen p< 4,notingthatthecase where p issmallis“perhapsthepracticallyinterestingcase”.
Perhapssomedemandpointsaremoreimportantthanothers.Plesnik[14] studiedthe priority k -centerproblem,inwhichtheeffectivedistancetoademand pointisincreasedinproportiontoitsspecifiedpriority.Plesnikapproximates thesymmetricversionwithinafactorof2.
Asymmetryin k -CenterVariants61
Charikaretal.[4]notethatadisadvantageofthestandard k -centerformulationisthatafewdistantclients, outliers,canforcecenterstobelocated inisolatedplaces.Theysuggestavariantoftheproblem,the k -centerproblem with outliersandforbiddencenters,whereasmallsubsetofclientscanbedenied service,andsomepointsareforbiddenfrombeingcenters.Charikaretal.gave a(bestpossible)3-approximationalgorithmforthesymmetricversionofthis problem.
Bhatiaetal.[3]consideredanetworkmodel,suchasacitystreetnetwork, inwhichthetraversaltimechangeasthedayprogresses.Thisisknownasthe k -centerproblemwithdynamicdistances :wewishtoassignthecenterssuchthat theobjectivecriteriaaremetatalltimes.
ResultsandOrganization
Table1givesanoverviewofthebestknownresultsforthevarious k -center problems.Inthispaperweexploreasymmetricvariantsthatarenotyetinthe literature.
Table1. Anoverviewoftheapproximationresultsfor k -centervariants. †β isthemaximumratioofanedge’sgreatestlengthtoitsshortestlength. ‡This isabicriteriaalgorithmusing k (1+3/(ν +1))centers. §For p< 4. ¶Thisisa bicriteriaalgorithmusing2k centers,for p ≤ n/k
Problem
Symmetric Asymmetric k -center 2[8] O (log ∗ k )[2] k -centerwithdynamicdistances 1+ β † [3] O (log ∗ n + ν ) ‡ [3] weighted k -center 3[9] O(log∗ n) Here p-neighbor k -center 3(2 §)[5] O(log∗ k) ¶ Here priority k -center 2[14] InapproximableHere
k -centerwithoutliersand 3[4] InapproximableHere forbiddencenters
k -suppliers 3[9] InapproximableHere
Section2containsthedefinitionsandnotationrequiredtodeveloptheresults.InSection3webrieflyreviewthealgorithmsofPanigrahyandVishwanathan[13],andArcher[2].Thetechniquesusedinthestandard k -center problemareoftenapplicabletothevariants.
Ourfirstresult,inSection4,isan O (log ∗ n)-approximationfortheasymmetricweighted k -centerproblem.InSection5wedevelopan O (log ∗ k )approximationfortheasymmetric p-neighbor k -centerproblem,for p ≤ n/k .Asnoted byKhulleretal.[12],thecasewhere p issmallisthemostinterestingcasein practice.Thisabicriteriaalgorithm,allowinganincreaseto2k centers,butit canbeturnedintoan O (log k )-approximationalgorithmusingonly k centers. Turningtohardness,weshowthattheasymmetricversionsofthe k -centerprob-
62IngeLiGørtzandAnthonyWirth
lemwithoutliers(andforbiddencenters),thepriority k -centerproblem,andthe k -supplierproblemare NP -hardtoapproximate(Section6).
2Definitions
Theinputtotheasymmetric k -centerproblemisadistancefunction d on ordered pairsofvertices—distancesareallowedtobeinfinite—andabound k onthe numberofcenters.
Definition2. Vertex c covers vertex v within r ,or cr -covers v ,if dcv ≤ r .We extendthisdefinitiontoaset C andaset A ifforevery a ∈ A thereisa c ∈ C suchthat c covers a within r .Oftenweabbreviate“1-covers”to“covers”.
Mostofthealgorithmsdonotinfactoperateongraphswithedgecosts.Rather, theyconsiderrestrictedgraphs,inwhichonlythoseedgeswithdistancelower thansomethresholdareincluded,andtheedgeshaveunitcost.Hochbaumand Shmoys[9]refertotheseasbottleneckgraphs.Sincetheoptimalvalueofthe coveringradiusmustbeoneofthe n2 distancevalues,manyalgorithmsessentiallyrunthroughasequenceofrestrictedgraphsofeverypossiblethreshold radiusinascendingorder.Thiscanbethoughtofas guessing theoptimalradius ROPT .Theapproachworksbecausethealgorithmeitherreturnsasolution, withinthespecifiedfactorofthecurrentthresholdradius,oritfails,inwhich case ROPT mustbegreaterthanthecurrentradius.
Definition3(RestrictedGraph Gr ). For r> 0,definetherestrictedgraph Gr ofthegraph G =(V,E ) tobethegraph Gr =(V,Er ),where Er = {(i,j ): dij ≤ r } andalledgeshaveunitcost.
Mostofthefollowingdefinitionsapplyto restricted graphs.
Definition4(PowerofGraphs). The tth powerofagraph G =(V,E ) isthe graph Gt =(V,E t ), t> 1,where E t isthesetofedgesbetweendistinctvertices thathaveapathofatmost t edgesbetweenthemin G.
Definition5. For i ∈ N define Γ + i (v )= {u ∈ G | (v,u) ∈ E i }∪{v },and Γi (v )= {u ∈ G | (u,v ) ∈ E i }∪{v } , i.e.intherestrictedgraphthereisapath oflengthatmost i from v to u,respectively u to v .
Noticethatinasymmetricgraph Γ + i (v )= Γi (v ).Weextendthisnotationto setssothat Γ + i (S )= {u ∈ G | u ∈ Γ + i (v )forsome v ∈ S } , with Γi (S )defined similarly.Weuse Γ + (v )and Γ (v )insteadof Γ + 1 (v )and Γ1 (v ).
Definition6. For i ∈ N define Υ + i (v )= Γ + i (v ) \ Γ + i 1 (v ),and Υi (v )= Γi (v ) \ Γi 1 (v ) , i.e.,thenodesforwhichthepathdistancefrom v isexactly i,andthe nodesforwhichthepathdistanceto v isexactly i,respectively.
Foraset S ,theextensionfollowsthepattern Υ + i (S )= Γ + i (S ) \ Γ + i 1 (S ).We use Υ + (v )and Υ (v )insteadof Υ + 1 (v )and Υ1 (v ).
Asymmetryin k -CenterVariants63
Definition7(CenterCapturingVertex(CCV)). Avertex v isa center capturingvertex (CCV)if Γ (v ) ⊆ Γ + (v ),i.e., v coverseveryvertexthatcovers v
Inthegraph GROPT theoptimumcenterthatcovers v mustliein Γ (v ).For aCCV v ,itliesin Γ + (v ),hencethename.Insymmetricgraphsallverticesare CCVsandthispropertyleadstothestandard2-approximation.
Definition8(DominatingSet). Givenagraph G =(V,E ),andaweight function w : V → Q+ onthevertices,findaminimumweightsubset D ⊆ V suchthateveryvertex v ∈ V iscoveredby D ,i.e., v ∈ Γ + (D ) forall v ∈ V
Definition9(SetCover). Givenauniverse U of n elements,acollection S = {S1 ,...,Sk } ofsubsetsof U ,andaweightfunction w : S→ Q+ ,finda minimumweightsub-collectionof S thatincludesallelementsof U .
The MaxCoverage problem,onaninstance U , S ,k ,issimilartotheSet Coverproblem:insteadoftryingto minimize thenumberofsetsusedwehavea bound onthenumberofsetswecanuse,andtheproblemisthentomaximizethe numberofelementscovered.TheDominatingSet,SetCover,andMaxCoverage problemsareall NP -complete.
3Asymmetric k -CenterReview
The O (log ∗ n)algorithmofPanigrahyandVishwanathan[13]hastwophases, the halve phase,sometimescalledthe reduce phase,andthe augment phase.As describedabove,thealgorithmguesses ROPT ,andworksintherestrictedgraph GROPT .InthehalvephasewefindaCCV v ,includeitinthesetofcenters,mark everyvertexin Γ + 2 (v )ascovered,andrepeatuntilnoCCVsremainunmarked. TheCCVpropertyensuresthat,aseachCCVisfound,therestofthegraphcan becoveredwithonefewercenter.Henceif k CCVsareobtained,theunmarked portionofthegraphcanbecoveredwith k = k k centers.Theauthors thenprovethatthisunmarkedportion,CCV-free,canbecoveredwithonly k /2 centersifweuseradius5insteadof1.Thatistosay, k /2centerssufficeinthe graph G5 ROPT
The k -centerproblemintherestrictedgraphisidenticaltothedominating setproblem.Thisisaspecialcaseofsetcoverinwhichthesetsarethe Γ + terms.Intheaugmentphase,thealgorithm recursivelyusesthegreedysetcover procedure.Sincetheoptimalcoverusesatmost k /2centers,thefirstcoverhas sizeatmost k 2 log 2n k
Thecentersinthisfirstcoverarethemselvescovered,usingthegreedy setcoverprocedure,thenthecentersinthesecondcover,andsoforth.After O (log ∗ n)iterationsthealgorithmfindsasetofatmost k verticesthat, togetherwiththeCCVs, O (log ∗ n)-coverstheunmarkedportion,sincetheoptimalsolutionhas k /2centers.Combiningthesewiththe k CCVs,wehave k centerscoveringthewholegraph.
64IngeLiGørtzandAnthonyWirth
Archer[2]presentstwo O (log ∗ k )algorithms,bothbuildingonthework in[13].Thealgorithmmoredirectlyconnectedwiththeearlierworkneverthelesshastwofundamentaldifferences.Firstly,inthereducephaseArchershows thattheCCV-freeportionofthegraphcanbecoveredwith2k /3centersand radius3.Secondly,heconstructsasetcover-likelinearprogramandsolvesthe relaxationtogetatotalof k fractional centersthatcovertheunmarkedvertices. Fromthesefractionalcenters,heobtainsa2-coveroftheunmarkedverticeswith k log k (integral)centers.Thesearetheseedfortheaugmentphase,whichthus producesasolutionwithan O (log ∗ k )approximationtotheoptimumradius.
Duringthepreparationofthefinalversionofthismanuscript,itwasannouncedthattheasymmetric k -centerproblemishardtoapproximatebetter than Ω (log ∗ n)[6,7],closingthegapwiththeupperbound.
4AsymmetricWeighted k -Center
Recalltheapplicationinwhichthecostsofdeliveryhubsvary.Inthissituation, ratherthanhavingarestrictiononthenumberofcentersused,eachvertexhas aweightandwehaveabudget W thatrestrictsthetotalweightofcentersused.
Definition10(Weighted k -Center). Givenaweightfunctiononthevertices, w : V → Q+ ,andabound W ∈ Q+ ,theproblemistofind S ⊆ V oftotalweight atmost W ,sothat S covers V withminimumradius.
HochbaumandShmoys[9]gavea3-approximationalgorithmforthesymmetricweightedversion,applyingtheirapproachforbottleneckproblems.We proposean O (log ∗ n)-approximationfortheasymmetricversion,basedonPanigrahyandVishwanathan’stechniquefortheunweightedproblem.Notethatin lightofthehardnessresultjustannounced[6,7],thisalgorithmisasymptoticallyoptimal.Anothervarianthasboththe k andthe W restrictions,butwe willnotexpandonthatproblemhere.
Firstabriefsketchofthealgorithm,whichworkswithrestrictedgraphs.In thereducephase,havingfoundaCCV, v ,wepickthelightestvertex u in Γ (v ) (whichmightbe v itself)asacenterinoursolution.Thenmarkeverythingin Γ + 3 (u)ascovered,andcontinuelookingforCCVs.Wecanshowthatthereexists a7-coveroftheunmarkedverticeswith totalweightlessthanhalfoptimum. Finallywerecursivelyapplyagreedyprocedureforweightedelements O (log ∗ n) times,similartotheoneusedforSetCover.Thetotalweightofcentersinour solutionsetisatmost W
Thefollowinglemmaaboutdigraphsisthekeytoourreducephaseandis analagoustoLemma4in[13]andLemma16in[2].
Lemma1(CoverofHalftheGraph’sWeight). Let G =(V,E ) beadigraphwithweightedvertices,butunitedgecosts.Thenthereisasubset S ⊆ V , w (S ) ≤ w (V )/2,suchthateveryvertexwithpositiveindegreeisreachableinat most3stepsfromsomevertexin S .
Asymmetryin k -CenterVariants65
Proof. Toconstructtheset S repeatthefollowing,totheextentpossible: Select avertexwithpositiveoutdegree,butifpossible select onewithindegreezero. Let v betheselectedvertexandcomparesets {v } and Γ + (v ) \{v }:addtheset ofsmallerweightto S andremove Γ + (v )from G.
Itisclearthattheweightof S isnomorethanhalftheweightof V .We mustnowshowthat S 3-coversallnon-orphanvertices—wecall x aparentof y if x ∈ Γ (y ).
Thechildrenof v areclearly1-covered.Assume v isnotin S (trivialotherwise):if v wasanorphaninitiallythenignoreit.If v isanorphanwhenselected, thensomeparentmusthavebeenremovedbytheselectionofagrandparent,so itis2-covered.
So v hasatleastoneparentwhenitisselected,implyingtherearenoorphan verticesatthattime.Thereforethesetsofparentsof v , S1 ,grandparentsof v , S2 ,andgreat-grandparentsof v , S3 ,arenotempty.Althoughthesesetsmight notbepairwisedisjoint,iftheycontainedanyof v ’schildren,then v wouldbe 3-covered.
After v isremoved,therearethreepossibilitiesfor S2 :(i)Somevertexin S3 is selected,removingpartof S2 ;(ii)Somevertexin S2 isselectedandremoved;(iii) Somevertexin S1 isselected,possiblymakingsome S2 verticeschildless.One oftheseevents must happen,since S1 and S2 arenon-empty.Asaconsequence, v is3-covered.
Henceforthcalltheverticesthat havenotyetbeencovered/marked active UsingLemma1wecanshowthatafterremovingtheCCVsfromthegraph, wecancovertheactivesetwithhalftheweightofanoptimumcoverifweare allowedtousedistance7insteadof1.
Lemma2(CoverofHalfOptimalWeight). Considerasubset A ⊆ V that hasacoverconsistingofverticesoftotalweight W ,butnoCCVs.Assumethere existsaset C1 that 3-coversexactly V \ A.Thenthereexistsasetofvertices S oftotalweight W/2 that,togetherwith C1 , 7-cover A.
Proof. Let U bethesubsetofoptimalcentersthatcover A.Wecall u ∈ U a near centerifitcanbereachedin4stepsfrom C1 ,anda far centerotherwise. Since C1 5-coversallofthenodescoveredbynearcenters,itsufficestochoose S to6-coverthefarcenters,sothat S will7-coverallthenodestheycover.
Defineanauxiliarygraph H onthe(optimal)centers U asfollows.Thereis anedgefrom x to y in H ifandonlyif x 2-covers y in G (and x = y ).Theidea istoshowthatanyfarcenterhaspositiveindegreein H .Asaresult,Lemma1 showsthereexistsaset S ∈ U with |S |≤ W/2suchthat S 3-coversthefar centersin H ,andthus6-coversthemin G
Let x beanyfarcenter.Since A containsnoCCVs,thereexists y suchthat y covers x,but x doesnotcover y .Since x ∈ Γ + 4 (C1 ), y ∈ Γ + 3 (C1 ),andthus y ∈ A (sinceeverythingnot3-coveredby C1 isin A).Thusthereexistsacenter z ∈ U ,whichisnot x,butmightbe y ,thatcovers y andtherefore2-covers x. Hence x haspositiveindegreeinthegraph H
Asweforeshadowed,wewillusethegreedyheuristictocompletethealgorithm.Wenowanalyzetheperformanceofthisheuristicinthecontextofthe dominatingsetprobleminnode-weightedgraphs.Allvertices V areavailableas potentialmembersofthedominatingset(i.e.centers),butweneedonlydominatetheactivevertices A.Theheuristicistoselectthemost efficient vertex: theonethatmaximizes w (A(v ))/w (v ),where A(v ) ≡ A ∩ Γ + (v ). Lemma3(GreedyAlgorithminWeightedDominatingSet). Let G = (V,E ), w : V → Q+ beaninstanceofthedominatingsetprobleminwhichaset A istobedominated.Also,let w ∗ betheweightofanoptimumsolutionforthis instance.Thegreedyalgorithmgivesanapproximationguaranteeof
Proof. Ineveryapplicationofthegreedyselectiontheremustbesomevertex v ∈ V forwhich
(A(v
otherwisenooptimumsolutionofweight w ∗ wouldexist.Thisiscertainlytrue ofthemostefficientvertex v ,somakeitacenterandmarkallthatitcovers, leaving A uncovered.Now, w (A )= w (A) w (A(v )) ≤ w (A) 1 w (v ) w ∗ ≤ w (A)exp w
After j steps,theremainingactivevertices, Aj ,satisfy
where vi isthe ithcenterpicked(greedily)and A0 istheoriginalactiveset.
Assumethataftersomenumberofsteps,say j ,therearestillsomeactive elements,buttheupperboundin(2)dropsbelow w ∗ .Thatistosay, j i=1 w (vi ) ≥ w ∗ ln(w (A0 )/w ∗ )
Beforewepickedthevertex vj wehad
j 1 i=1 w (vi ) ≤ w ∗ ln(w (A0 )/w ∗ ) , andso, j i=1 w (vi ) ≤ w ∗ + w ∗ ln(w (A0 )/w ∗ ) , because(1)tellsusthat w (vi )isnogreaterthan w ∗ .Tocovertheremainder, Aj ,wejustuse Aj itself,atacostofatmost w ∗ .Hencethetotalweightofthe solutionisatmost w ∗ (2+ln(w (A0 )/w ∗ )).
Ontheotherhand,iftheupperboundon w (Aj )neverdropsbelow w ∗ before Aj becomesempty,thenwehaveasolutionofweightatmost w ∗ ln(w (A0 )/w ∗ ).
Asymmetryin k -CenterVariants67
Wenowshowthatthistradeoffbetweencoveringradiusandoptimalcover sizeleadstoan O (log ∗ n)approximation.
Lemma4(RecursiveSetCover). Given A ⊆ V ,suchthat A hasacoverof weight W ,andaset C1 ∈ V thatcovers V \ A,wecanfindinpolynomialtimea setofverticesoftotalweightatmost 2W that,togetherwith C1 ,cover A (and hence V )withinaradiusof O (log ∗ n)
Proof. Ourfirstattemptatasolution, S0 ,isallverticesofweightnomorethan W :onlytheseverticescouldbeintheoptimumcenterset.Theirtotalweightis atmost nW .Since C1 covers S0 \ A,consider A0 = S0 ∩ A,whichhasacover ofsize W .Lemma3showsthatthegreedyalgorithmresultsinaset S1 that covers A0 ,andhasweight w (S1 ) ≤ O W log Wn W = O (W log n)
Set C1 covers S1 \ A,soweneedonlyconsider A1 = S1 ∩ A,andsoforth.Atthe ithiterationwehave: w (Si ) ≤ O (W log(w (Si 1 )/W ))andhencebyinduction atmost O (W log(i) n).Thusafterlog∗ n iterationstheweightofoursolutionset fallsto2W .
Allthealgorithmictoolscanbeassembledtoformanapproximationalgorithm. Theorem1(ApproximationofWeighted k -Center). Wecanapproximate theweighted k -centerproblemwithinfactor O (log ∗ n) inpolynomialtime.
Proof. Guesstheoptimumradius, ROPT ,andworkintherestrictedgraph GROPT Initially,theactiveset A is V .Repeatthefollowingasmanytimesaspossible: PickCCV v in A,addthelightestvertex u in Γ (v )tooursolutionsetofcenters and,removetheset Γ + 3 (u)from A.Since v iscoveredbyanoptimumcenterin Γ + (v ), u isnoheavierthanthisoptimumcenter,and Γ + 3 (u)includeseverything coveredbytheoptimumcenter.
Let C1 bethecenterschoseninthisfirstphase.Weknowtheremainderof thegraph, A,hasacoveroftotalweight W = W w (C1 ),becauseofourchoices basedonCCVandweight.
Lemma2showsthatwecancovertheremaininguncoveredverticeswith weightnomorethan W /2ifweusedistance7.Solettheactiveset A be V \ Γ + 7 (C1 ),andrecursivelyapplythegreedyalgorithmasdescribedinthe proofofLemma4onthegraph G7 ROPT .Asaresult,wehaveasetofsize W that covers A withinradius O (log ∗ n).
5Asymmetric p-Neighbor k -Center
Imaginethatwewishtolocate k facilitiesatsuchthatthemaximumdistance ofademandpointfromits pth -closestfacilityisminimized.Asaconsequence, failuresin p 1facilitiesdonotbringdownthenetwork.
68IngeLiGørtzandAnthonyWirth
Definition11(Asymmetric p-Neighbor k -CenterProblem). Let dp (S,v ) denotethedistancefromthe pth closestvertexin S to v .Theproblemistofind asubset S ofatmost k verticesthatminimizes
(S,v )
Weshowthatwecanapproximatetheasymmetric p-neighbor k -centerproblemwithinafactorof O (log ∗ k )ifweallowourselvestouse2k centers.Our algorithmisrestrictedtothecase p ≤ n/k ,butthisisreasonableas p should notbetoolarge[12].
Weusethesametechniquesasusual,includingrestrictedgraphs,butinthe augmentphaseweusethegreedyalgorithmforthe ConstrainedSetMulticover problem[15].Thatis,eachelement, e,needstobecovered re times,buteach setcanbepickedatmostonce.The p-neighbor k -centerproblemhas re = p forall e.Wesaythatanelement e is alive ifitoccursinfewerthan p sets chosensofar.Thegreedyheuristicisto pickthesetthatcoversthemostlive elements.Itcanbeshownthatthisalgorithmachievesanapproximationfactor of Hn = O (log n)[15].Howeverthefollowingresultismoreappropriatetoour application.
Lemma5(GreedyConstrainedSetMulticover). Let k betheoptimum solutiontotheConstrainedSetMulticoverproblem.Thegreedyalgorithmgives approximationguarantee O (log(np/k ))
Proof. ThesamekindofaveragingargumentusedforstandardSetCovershows thatthegreedychoiceofasetreducesthetotalnumberofunmarkedelement copiesbyafactor1 1/k .Soafter i stepsthenumberofcopiesofelements yettobecoveredis np(1 1/k )i ≤ np(e 1/k )i .Henceafter k ln(np/k )stepsthe numberofuncoveredcopiesofelementsisatmost k .Anaivecoveroftheselast k elementcopiesleadstothetotalnumberofsetsbeing k + k ln(np/k ).
If p ≤ n/k thisgreedyalgorithmgivesanapproximationfactorof O (log(n/k )). Applyingthestandardrecursiveapproachin[13],whichworksinthe p-neighbor case,wecanachievean O (log n)approximationwith k centers,or O (log ∗ n)with 2k centers.Wecanlowertheapproximationguaranteeto O (log ∗ k ),with2k centers,usingArcher’sLP-basedpriming.FirstsolvetheLPfortheconstrainedset multicoverproblem.Inthesolutioneachvertexiscoveredbyanamount p of fractionalcenters,outofatotalof k .Wecannowusethegreedysetcoveralgorithmtogetaninitialsetof k 2 ln k centersthat2-coverseveryvertexinthe activesetwithatleast p centers.Repeatedlyapplyingthegreedyprocedurefor constrainedsetmulticover,thistimefor(log∗ k +1)iterations,weget2k centers thatcoverallactiveverticeswithin O (log ∗ k ).Alternatively,wecouldcarryout O (log k )iterationsandsticktojust k centers.
6InapproximabilityResults
Inthissectionwegiveinapproximabilityresultsfortheasymmetricversions ofthe k -centerproblemwithoutliers,thepriority k -centerproblem,andthe
Asymmetryin k -CenterVariants69
k -supplierproblem.Theseproblemsalladmitconstantfactorapproximation algorithmsinthesymmetriccase.
Asymmetric k -CenterwithOutliers
Definition12(k -CenterwithOutliersandForbiddenCenters). Finda set S ⊆ C ,where C isthesetofverticesallowedtobecenters,suchthat |S |≤ k and S coversatleast p nodes,withminimumradius.
Theorem2. Foranypolynomialtimecomputablefunction α(n),theasymmetric k -centerproblemwithoutliers(andforbiddencenters)cannotbeapproximatedwithinafactorof α(n),unless P = NP
Proof. Wereduceinstance U, S ,k ofMaxCoveragetoourproblem.Construct vertexsets A and B sothatforeachset S ∈S thereis vS ∈ A,andforeach element e ∈ U thereis ve ∈ B .Fromeveryvertex vS ∈ A,createanedgeofunit lengthtovertex ve ∈ B if e ∈ S .
Let p = |B | + k ,sothatifwefind k centersthatcover p verticeswithinany finitedistance,we must havefound k verticesin A thatcoverall |B | vertices. HencewehavesolvedtheinstanceofMaxCoveragewhichisan NP -complete problem.
Notethattheproofneverreliedonthefactthatthe B verticeswereforbidden frombeingcenters(setting p to |B | + k ensuredthis).
AsymmetricPriority k -Center
Definition13(Priority k -Center). Givenapriorityfunction p : V → Q+ onthevertices,find S ⊆ V , |S |≤ k ,thatminimizesRsothatforevery v ∈ V thereexistsacenter c ∈ S forwhich pv dcv ≤ R
Theorem3. Foranypolynomialtimecomputablefunction α(n),theasymmetric k -centerproblemwithprioritiescannotbeapproximatedwithinafactorof α(n),unless P = NP .
Proof. Theconstructionofthesets A and B isthesimilartotheproofofTheorem2,exceptthatwereducefromSetCover.Thistimemaketheset A a completedigraph,withedgesoflength ,aswellastheunitlengthset-element edgesfrom A to B .Givethenodesinset A priority1andthenodesinset B priority .Anoptimalsolutiontothepriority k -centerproblemis k centersin A andaradiusof ,whichcoverseveryvertex.Thisimpliesthatthe k centers cover(intheSetCoversense)alltheelementsin B .If k <k centerswerechosen from A and k k centerswerechosenfrom B instead,wecouldtriviallyconvert thistoasolutionchoosing k centersfrom A
Anynon-optimalsolutionrequiresaradiusofatleast 2 + ,asthiswould involvecoveringsome B vertexbysteppingfroman A centerthroughanother A vertex.Thereforeanyalgorithmwithapproximationguarantee +1 ε or betterwouldsolveSetCover.Wecanmake anyfunctionwelikeandtheresult follows.
70IngeLiGørtzandAnthonyWirth
Asymmetric k -Supplier
Definition14(k -Supplier). Givenasetofsuppliers Σ andasetofcustomers C ,findasubset S ⊆ Σ thatminimizes R suchthat S covers C within R
Theorem4. Foranypolynomialtimecomputablefunction α(n),theasymmetric k -supplierproblemcannotbeapproximatedwithinafactorof α(n),unless P = NP
Proof. ByareductionfromtheMaxCoverageproblemsimilartotheproofof Theorem2.
Acknowledgements
TheauthorswouldliketothankMosesCharikarandthereviewers.
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