observethatwhenthinkingaboutadrawingwheretheedgesaredrawnmostlyvertica - PDF document

observethatwhenthinkingaboutadrawingwheretheedgesaredrawnmostlyvertical,wewillusuallyalsohavealownumberofcrossings.Furthermore,edgestendtocrossonlyonaverylocalscale(i.e.,edgeswillusuallynotcrossoveralargehorizontaldistance),increasingthedrawing'sreadability[21].Hence,perhapsthecombinationofmaximumverticalityandlowcrossingnumberleadsto(qualitatively)betterdrawingsthanthetraditionalminimumcrossingnumberinconjunctionwithhighverticality.Theassumptionthathighverticalityleadstofewcrossingsandgooddraw-ingsisalsosupportedbythefollowingobservation:Thebarycenterheuristicisoneoftheearliest,andstillprobablythemostcommonheuristictoquicklysolvethelayeredcrossingminimizationprobleminpractice,especiallyforlarge-scalegraphs.Yet,theheuristicdoesactuallynotactivelytrytominimizecrossings,butiterativelydecidesonpositionspofverticesonlayeri,suchthatpliesat(orcloseto)thebarycenterofthepositionsofitsadjacentnodesonleveli�1.So,theheuristicisinfactmainlytryingtooptimizeourverticalityproblem!Itscrossingminimizationpropertiesariseonlyinthewakeofthisoptimizationgoal.Aswewillbrie ydiscussbelow,ourproblemisaspecialformofanorderingproblem,whichalsoarisesinareasunrelatedtographdrawing.Assuch,wecalltheproblemMulti-levelVerticalOrdering(MLVO).Whenspecicallytalkingaboutthegraphdrawingapplication,i.e.,ndingorderingsofthenodesontheirlevelssuchthattheedgesaredrawnasverticalaspossible(seeaprecisedenitionbelow)weusethetermMulti-levelVerticalityOptimization,which,nicelyenough,givesrisetothesameabbreviation.Aswewillsee,MLVOisanaturalquadraticorderingproblem(QOP).OfcourseMLVOisNP-hard(seeAppendixA.1)andcloselyrelatedtothetradi-tionalproblemofmulti-levelcrossingminimization(MLCM),whereoneseeksnodeorderssuchthatthenumberofcrossingsinmulti-leveldrawingsismini-mized.MLCMhasreceivedalotofattentionnotonlywithinthegraphdraw-ingcommunity,butincombinatorialoptimizationingeneral;see,e.g.,[2,5]foroverviewsonstrongheuristicsandexactalgorithmstotackletheproblem.MLVOisalsorelatedtotheproblemofmulti-levelplanarization(MLP)[19,10],i.e.,ndnodeorderswhichminimizethenumberofedgesthathavetoberemovedinordertoobtainaplanar(sub)drawing.ThisproblemhasbeenproposedasapossiblesubstituteforMLCM,suggestingthatitcanresultinmoreaestheticallypleasingdrawings.1.1FocusandContributionThefocusofthispaperistopresenttheconceptandspecicationofMLVOinitsnativegraphdrawingsetting,discussitsrelativemeritsandchallenges,showitssolvabilityviavariousalgorithmicstrategies,andgiveanoverviewonpossiblefurtherextensions.Therefore,itisbeyondthescopeofthispapertogivein-depthdetailsofinnerworkingoftherathercomplexexactalgorithmstotackletheproblem.Forsuchadiscussionsee[4],whereweconsiderILP-,QP-,andSDP-basedalgorithmstotacklethebaseproblemofMLVO.Althoughgraphdrawingisthemain(andmostdeveloped)applicationarea,MLVOcan2 (a)linearcost (b)proper,narrow (c)proper,wide (d)non-prop.,wideFig.1.Exampledrawingsregardingverticalitymaximization:(a)equivalentqualitywithrespecttoalinearobjectivefunction,(b){(d)dierentdrawingparadigms,cf.text.Originalnodesaredrawnaslargegraycircles,LEDsasblacksmallcircles,PDs(ontheemptygridpoints)areomittedforreadability.linkedbilevelQOPs(oneforeachpairofadjacentlevels).WewillseethatMLVOcannotonlybeappliedinsuchasetting(resultinginproperdrawings),butalsodirectlytonon-propergraphs(resultinginnon-properdrawings):thisgivesriseto\true"multi-levelQOPsasalllevelscandirectlyinteractwitheachother.2.1VerticalityWedenethecolloquialtermverticalityviaitsinverse,non-verticality:Thenon-verticalityd(e)ofastraight-lineedgeeisthesquareofthedierenceinthehorizontalcoordinatesofitsendnodes.Then,d(E):=Pe2Ed(e)denotestheoverallnon-verticalityofasolution.Usingonlythisnotion,wecouldarbitrarilyoptimizeadrawingbyscalingthehorizontalcoordinates.Henceweconsidergriddrawings,i.e.,thenodes'positionsaremappedtointegralcoordinates,therebyrelatingverticalitytothedrawing'swidth.Clearly,weonlyconsideradjacentintegralsforthey-coordinates.Itremainstoarguewhynon-verticalityhastobeaquadraticterm:assumewewouldonlyconsideralinearfunction,thenevenasmallexampleastheonedepictedinFig.1(a)wouldresultinmultiplesolutionsthatareequivalentw.r.t.theirobjectivevalues,eventhoughthebottomoneisclearlypreferablefromthereadabilitypointofview.Intuitively,weprefermultipleslightlynon-vertical4 edges,overfewbutverynon-verticaledges.Infact,thisargumentbringsourmodelinlinewiththeargumentofobservingcrossingsonlyonalocalscale.2.2ProperDrawingSchemeWecanconsidertwodistinctalignmentschemes,duetothefactthatthenodepartitionsV0ihavedierentcardinalities.Let!0:=max1ipjV0ijdenotethewidthofthewidestlevel.Inthenarrowalignmentschemes,werequirethenodesonthelevelstolieondirectlyadjacentx-coordinates(Fig.1(b)).Then,wewouldusuallyliketocenterthedistinctlevelsw.r.t.eachother,i.e,alevelimayonlyusethex-coordinatesf0i;:::;0i+jV0ij�1g,withthelevel'swidthoset0i:=b(!0�jV0ij)=2c.Thebenetofthisalignmentschemeisthatasimplelinearorderofthenodesperlayeralreadyfullydescribesthesolution.Yet,notethatinmostcasessuchanalignmentschemewillnotresultinaestheticallypleasingdrawings.Inthewidealignmentscheme(Fig.1(c)),nodesarenotrestrictedtolieonhorizontallyneighboringgridcoordinates.Inordertomodelthis,weonlyneedtoexpandthegraphbyaddingpositionaldummynodes(PDs)toeachlevelsuchthatalllevelshave!0manynodes.AllPDshavedegree0.Inthefollowing,wewillsimplyconsiderany(proper)levelgraphG(G0,respectively),whichmayormaynotbeaugmentedwithPDs.Ingraphdrawingpractice,wewillusuallyonlyusethiswidealignmentscheme.Monotonousdrawings.Consideringdrawingsoptimalw.r.t.MLVO,wemaywanttoforceanadditionalmonotonicityproperty.WithintheSugiyamaframework,eachedgeisdrawnusingonlystronglymonotonouslyincreasingy-coordinates.Itwouldbenicetohaveasimilarpropertyalongthex-axisoveralledgesegmentscorrespondingtoanoriginaledge.Wesayadrawingismonotonous,ifalloriginaledgesareweaklymonotonousalongthex-axis.Moreformally,lete=(u;v)2Ebeanyedgeinthe(non-proper)levelgraphG,e1=(u=u0;u1);e2=(u1;u2);:::;ek=(uk�1;uk=v)thecorrespondingchainofedgesinG0,andx:V0!Nthemappingofnodestox-coordinatesinthenaldrawing.Thenadrawingismonotonous,ifx(u)()x(v)impliesx(ui)()x(ui+1)forall0ik.Eventhoughthismightbecounterintuitiveatrst,anoptimalMLVOso-lutiondoesnotinducethisproperty.Wemay,however,explicitlyaskforthispropertytohold,givingrisetothemonotonousMLVOproblem.3Non-ProperDrawingSchemeWhenlookingattypicalSugiyama-styledrawings,weoftenobservethatLEDs|eventhoughtheyareneverexplicitlydrawn|aregiventoomuchspace:Objec-tively,itisunreasonableforLEDstorequireasmuchhorizontalspaceasarealnode.Therefore,currentdrawingalgorithmstryhardto\bundle"multiplelong5 3.1HypotheticalandshiftedroutingThey-andx-coordinatesofthenodesarexedbythelayering(`)andnodeorderperlayer,respectively.Asageneralidea|calledthehypotheticalrouting|wewanttodraweachedgeverticallyuptotheleveldirectlybelowthetargetnode.Onlythere,theedgebendstobedrawnasalinewiththecomputednon-verticality.Clearly,thereareproblemswiththissimpleconcept:Firstly,routingedgesstrictlyverticalmayrequiretodrawthemthroughothernodes.Secondly,verticalsegmentsofmultipleedgeswouldcoincide.Toavoidtheseissues,wehavetorelaxthehypotheticalroutingsuchthatwerouteanedgee=(u;v)vertically\closeto"thex-coordinateofthesourcenode(i.e.,shiftedbysomesmalls(e)).Moreformally,theedgestartsatthecoordinate(x(u);`(u)),hasarstbendpointat(x(u)s(e);`(u)+1)shiftingtheedgeeithertotheleftortotheright(dependingontheedge'soveralldirection),andisroutedverticallyuntilthepoint(x(u)s(e);`(v)�1)whereitbendstogostraighttotheendpoint(x(v);`(v)).Forshortedgesors(e)=0,somebendpointsmayvanishintheobviousway.Whens(e)isassumedsmallenough,theoverallnon-verticalityofthisroutingisroughlyequivalenttotheverticalityachievedbythehypotheticalrouting.Observethatverticaledges(i.e.,x(u)=x(v))aresomehowspecialasitisnotperseclear,whethers(e)shouldbeaddedorsubtracted;wewilldiscussthisuncertaintylater.Ouroverallgoalisthatthecrossingsinducedbythisshiftedroutingsatisfythefollowingproperties:(P1)adjacentedgesdonotcross,allotheredgepairscrossatmostonce;(P2)averticaledgee1mayonlycrossanotheredgee2(butisnotrequiredto!)whenexactlyoneendnodeofe2isverticallybetweentheendnodesofe1;and(P3)twonon-verticaledgescrossexactlyiftheirhypotheticalroutingscross.Inordertoachievetheseproperties,theshiftvaluess(e)fortheedgeshavetobechosencarefully.Monotonousdrawings.Clearly,thehypotheticaldrawingismonotonousinthex-andy-coordinates,aswellasstrictlymonotonous(i.e.,intheirgeneraldirection).Duetoourshifting,thisproperty(necessarily)getsslightlyviolatedforedgesdrawnstrictlyverticalinthehypotheticalrouting.Wemaysayadrawingis-monotonousforsome,ifedgeswithidenticalsourceandtargetx-coordinatesdeviatefromthiscoordinatebyatmostatanypoint,andallotheredgesaredrawnmonotonously.3.2ComputingShiftsLetV�Vdenotetheoriginalvertices(incontrasttopossiblePDs).Largery(x)coordinatesarehigher(moreright,respectively)inthedrawing.Weit-erativelyconsiderallnodesv2V�,indecreasingorderoftheiry-coordinate.Foralledgese2Ev:=f(v;u)2E:`(u)�`(v)gthathavevastheirsource7 therespectivel,r-partitionsoftheverticaledges,i.e.,thepartitionsofE=vintoE=;lv;E=;rv,forallv2V�.Lemma2.Fixingalll,r-partitions,theabovedrawingalgorithmrequirestheminimumpossiblenumberofcrossings.Yet,evenwhengiventhenodeordersperlevel,obtainingl,r-partitionsthatleadtotheoverallminimumnumberofcrossingsisNP-hard.Proof.Therstpartfollowsfromthealgorithmicdescription.TheNP-hardnessfollowsfromthefactthatalreadyasinglecolumnofverticallyarrangedverticesconstitutestheNP-hardxedlinearcrossingnumberproblem[17].Inpractice,thepartitionproblemisusuallynotcritical:thenumberofcross-ingsbetweenpairsofverticaledgesisusuallydominatedbythecrossingsin-volvingnon-verticaledges.Infact,inourimplementationwesettleonaverysimple,yetseeminglysucient,heuristic:duringthealgorithm,wegreedilypickthesidewheretheedgeattainsthesmallerlabel;webreaktiesbyclassifyingedgeswhosesourcenodevisontheleft(right)halfofthedrawingasE=;lv(E=;rv,resp.).Thistiebreakingisreasonablesince,consideringanodevontheleftsideofthedrawing,itwillusuallyhavemoreadjacentnodestoitsrightthantoitsleftside,andhencethedecisionusuallyleadstofewercrossings.Basedonthefactthatallsortingisdoneonintegralvalues,wecanconclude:Theorem1.Theabovedrawingalgorithmgeneratesanon-properdrawingofalevelgraphG=(V;E)withspeciednodeordersperlevelinO(jVj+jEj)time.Theedges'routingsaremonotonousintheiry-coordinates,-monotonousintheirx-coordinate,andrealizetheminimalnumberofcrossings(w.r.t.thegivennodeordersandl,r-partitions).4SolvingMLVO4.1BarycenterandMedianWealreadynotedintheintroductionthattraditionalMLCMheuristicsinfactoftenoptimizethedrawings'verticality(inanarrowalignmentschemesetting).Inparticular,wecanusethetraditionalapproachofcomputingthebarycenterormedianforthenodes,byonlylookingatxedpositionsofthenodesonelevelbelow/above(incaseofproperdrawings),oronanylevelbelow/above(incaseofnon-properdrawings),andsortthemaccordingly.Iteratingthisprocedureforalllayersbothintheupwardanddownwarddirection(i.e.,alternatinglyconsiderthelevelsbeloworabove)untilnomoreimprovementispossibleminimizesthenumberofcrossingsonlyindirectly,buttheedges'verticalitydirectly.Whenconsideringthewidealignmentscheme,weobservethatwecannotcomputereasonablevaluesforPDsastheyhavenoincidentedges.Therefore,werstonlycomputethebarycenter/medianfortheoriginalnodes|wecallthesevaluesthedesiredlocationsofthenodes|andsortthemaccordingly.Then,wetrytodispersethePDs(necessarytoachievethelayerwidth!0)intothislist9 feasiblesolutions,i.e.,upperbounds.Weiteratethisprocessuntilthegapscoin-cide(afterroundingthelowerboundtothenextintegerabove),oraniterationlimitisreached.MLVOafterMLCM(MLVAC).OurMLVOSDPcannotonlybeuseddirectlyaftertheSugiyama'srststage,butwecanalsoapplyitafterasecondstagecrossingminimization,i.e.,aftersolvinganMLCMproblem.Byxingtheorderoftheoriginalnodes(non-PDs),theSDPbecomesanexactquadraticcompactorforSugiyama'sthirdstage.Suchaxingcanbeachievedeitherbydroppingthexedvariablesaltogetherandcorrespondingmodicationstotheconstraintmatrix,orbyintroducingequalityconstraintsontherespectivevariables.Inourexperiments,weusedthelatterapproachduetocodesimplicity.Implementingthereductionstrategywouldassuminglyleadtofurtherimprovedrunningtimes.Letp:V!f0;1;:::;!gbetherelativepositionfunction,wherep(u)=0meansthattherelativepositionofnodeuisnotxed.Weaskforthefollowingconstrainttoholdfortwonodesu_2Vmwithp(u)&#xv]TJ;&#x/F14;&#x 9.9;ئ ;&#xTf 1;.70; 0 ;&#xTd [;0;p(v)&#xv]TJ;&#x/F14;&#x 9.9;ئ ;&#xTf 1;.70; 0 ;&#xTd [;0yuv=1;ifp(u)p(v),yuv=�1;ifp(u)&#x-278;p(v).(1)Wecanfurtherstrengthenthesemideniterelaxationbyadding�1linear-quadraticconstraintsthatwegetfrommultiplying(1)withanarbitraryorderingvariableyst;s_2Vn.WealsohavetoadapttheSDPheuristic.Wextheorderingofthe\real"nodesandLEDsbeforehyperplaneroundingandthenonlyallowto ipsignsofvariablesinvolvingPDs.5ComputationalExperienceWecomparetherelativebenetsofthedierentdrawingschemesandsolutionmethodsdiscussedaboveandshowcasetheirvisualresults.ThereforeweapplytheexactSDPapproachandtheheuristicsproposedintheprevioussectiontosolveMLVOonavarietyoftestsets.Theaimistoinvestigatetheirgeneralap-plicability,notsomuchanin-depthanalysisandmeritevaluation,whichwouldbebeyondthescopeofthispaper.AllcomputationswereconductedonanIn-telXeonE5160(Dual-Core)with24GBRAM,runningDebian5.0in32bitmode.TheSDPalgorithmrunsontopofMatLab7.7,whereastheheuristicsareimplementedinC++.TheSDPapproachleavessomeroomforfurtherin-crementalimprovementaswerestrictthenumberofiterationstocontroltheoverallcomputationaleort.Fortheheuristic,wegivethetotalrunningtimeandbestfoundsolution,considering500independentruns.Weobservethatwhile(forlargergraphs)thisisbenecialtofewerruns,therearenearlynomoreimprovementsinthesolutionqualitywhenfurtherincreasingthisnumber.Allgraphsconsideredinthissection(includingtheiroptimalsolutions,whereavailable),aswellasanimplementationofthenon-planardrawingstyle,areonlineathttp://www.ae.uni-jena.de/Research_Pubs/MLVO.html.11 SDP Heuristic Instance dtime d50d500time500 Proper Polyt. Octahedron 239+50:03:28 2442440.70 Dodecahedron 1815+8129:55:48 1837183410.78 Cube4 5279+12180:49:47 5364536033.74 Gr.viz unix 58+51:19:41 74691.73 world 331+9554:30:21 48647914.84 prole 876+16995:45:58 96295921.57 Other MS88 155+20:52:17 1571572.62 Worldcup86 349+261:44:46 3683562.60 Worldcup02 385+157:19:38 4053996.44 SM96-full 658+36137:21:07 80975839.78 Non-proper Gr.viz unix 30+30:10:11 34330.28 world 103+70:43:50 1141090.62 prole 254+53:11:43 2662601.87 Other Worldcup86 113+30:31:30 1161160.44 Worldcup02 150+10:36:42 1561510.73 SM96-full 408+92:16:06 4354213.54 Table2.Dierentapproachesforproperandnon-properMLVOwithwidealignmentscheme.Thetimeissuitablygiveneitherinsecondsorashh:mm:ss.d=X+YgivesthenallowerboundXandupperboundX+Yoftheverticality.Theheuristicusesarandominitialorderandbothlocaloptimizationschemes(startingwith2-opt)alternatingly.Wegivethebestresultsafter50and500independentruns;thetimeisspeciedasthetotalfor500independentruns.Motivatedbytheseresults,wenowinvestigateouralgorithmsontwowell-knownlargerbenchmarksets,whichwerealsoconsideredinthecontextof(ex-act)multi-levelcrossingminimization[5].TheRomegraphs[7]contain11,528instanceswith10{100verticesand,althoughoriginallyundirected,canbeunam-biguouslyinterpretedasdirectedacyclicgraphs,asproposedin[8].TheNorthDAGs[6]contain1,158DAGs,with10{99arcs.Consistentwith[5],weconsidertwodierentwaysoflayeringthegraphsofbothbenchmarksets:theoptimalLP-basedalgorithmby[11]andthelayeringresultingfromcompactinganup-wardplanarization[3].BothyieldsimilarresultsintermsofMLVOrunningtimeandsolvability.Ourmainndingisthattheoverallobservationw.r.t.theheuristicvariantshold.Yet,asthelayeringsintroducemanymoreLEDsthanthegraphsconsideredbefore,theadvantageofnotrequiringLEDsbecomesevenmorepronounced:consideringthelargestgraphsoftheNorthDAGs(Romegraphs)withoriginallymorethan90edges(nodes),asinglerunoftheheuristicrequiresonly6ms(2ms)onaverage,whereasthepropergraphsrequire1.8sec(0.8sec,resp.).Similarly,theSDPapproachisapplicabletoallRomeandnearlyallNorthgraphs(98%)inthenon-propersetting,astheapproachworkswellupto5000.Ityieldsaveragegapsrangingfrom5%to80%withgrowing13 MLCM MLVO MLVAC Instance zdtime zdtime d(z)time Polyt. Octahedron 8026410.66 81261+10:02:37 243+10:11:49 Dodecahedron 393+130964:40:09 3993051+273:31:58 1851+157:57:00 Cube4 1192+365947:10:19 12476336+867:57:46 5414+3236:23:34 Gr.viz unix 01410.25 71110:04:27 86+51:01:46 world 468471:13:49 83620+416:33:10 459+2917:46:48 prole 3727670:53:34 751303+97:09:51 1363+38178:54:16 Other MS88 913002.79 1092490:01:27 154+40:48:26 Worldcup86 4976225.3 725590:05:43 506+52:01:36 Worldcup02 457900:01:33 63501+11:24:56 520+93:17:16 SM96-full 16214910:53:29 2221212+138:47:37 711+6761:30:45 Table3.ComparingProperMLVOwithnarrowalignmentschemewithMLCM,andcombiningthemtoMLVAC.Thecolumnszanddgivetheoptimalsolutions(ornalbounds)ofMLCMandMLVO,respectively.Thecolumnsz,dgivethecrossingnumberandnon-verticalityofthefoundsolution.d(z)givesboundsontheoptimalnon-verticalitywithassuredminimalcrossingnumber.Thetimeissuitablygiveneitherinsecondsorashh:mm:ss.15 6.3NodesizesInmanyreal-worldscenarios,itcanbeinterestingtoconsidernodesofvary-ingsize.Before,anynoderequiredexactlyonegridpoint;generally,wemayintroducenodesrequiringdxdygridpoints.Ahorizontalstretchiseasytoincorporate:whenconsideringtheabsolutegridpositionofanodewenotonlycomputethenumberofnodestoitsleft,butthesumoftheirhorizontalstretches.Toincorporateverticalstretches,wecopythenodeonallitsrespectivelevelsandconnectthemfromleveltolevelwithdummyedges.Now,weonlygeneratesolutionswherethesedummyedgesarestrictlyverticalandnotcrossed,bothofwhichcanbeachievedintheSDPstraight-forwardly:Fortheformer,wesimplyaddcorrespondingequalities.Forthelatter,letubeanodeverticallystretchedbetweenlayers`0and`1,x(u)itsx-coordinate,and(w0;w1)anyotheredgewith`0`(w1)`1(i.e.,apotentiallycrossingedge).Werequire[x(w0)�x(u)][x(w1)�x(u)]0:7ConclusionsandFurtherThoughtsWesuggestedtheconceptofverticalityasanovelexplicitoptimizationgoal.Weshowedrstapproachestotackletheprobleminpracticeandderivedanewdrawingstylebasedonthisconcept;thelatterallowstomeaningfullyabon-donthegraph-enlargingedgesubdivisionintrinsictothetraditionalSugiyamascheme.Ourconceptoersinterestingfurthertopicsforresearch:{Inourtestset,verticality-wiseoptimaldrawingsaretypicallyverygoodw.r.t.thecrossingnumber,andviceversa.Yet,itisanopen(graphtheoretic)question,howmuchthesetwomeasurescandeviateintheirrespectivelyoptimaldrawings.Inotherwords,howbad(intermsofcrossingnumber)canaverticality-wiseoptimaldrawingbecome,andviceversa?{Itseemsworthwhiletoinvestigatefurther,moreinvolved,algorithmsthatclosethegapbetweenoursimpleheuristicsandthecomputationallyexpen-siveSDPapproach.Can,e.g.,sifting-basedalgorithmslike[2]beadoptedtoecientlyworkforverticalityoptimization?Quitegenerally,MLVOseemstobeaninterestingplaygroundtostudyhowtoadopttheextendedresearchonMLCMalgorithmstoanewbutrelatedeld.{Intentionally,thisarticleleavesonecentralquestionunanswered:Isaverticality-optimaldrawing\better"intermsofperceptionthanacrossingminimumdrawing.Answeringthisquestiongoeswellbeyondthescopeofthispaper:ontheonehanditwouldrequireawell-constructeduserstudy.Ontheotherhand,suchastudyisnotyetfeasible:Asnotedabove,wearestillmissingal-gorithmstoobtainpracticallynear-optimalsolutionsforgraphstoolargeforourSDP.Onlythen,wecancomparesuchresultsto(near-)optimalMLCMsolutionsinameaningfulway.17 21.H.C.Purchase.Whichaesthetichasthegreatesteectonhumanunderstanding?InGD'1997,LNCS1353,pages248{259,1997.22.F.ShiehandC.McCreary.Directedgraphsdrawingbyclan-baseddecomposition.InGD'95,LNCS1027,pages472{482,1996.23.K.Sugiyama,S.Tagawa,andM.Toda.Methodsforvisualunderstandingofhierarchicalsystemstructures.IEEETrans.Sys.,Man,Cyb.,11(2):109{125,1981.19 Thevariablesshallbe1ifuisleftofvand�1otherwise.Fornotationalsim-plicity,wealsousetheshorthandyvu:=1�yuvforu_.Itiswell-knownthatfeasibleorderingscanbedescribedvia3-cycleinequalities�1yuv+yvw�yuw1;8u;v;w2Vi;1ip;u__:(2)Takingthevectorycollectingtheyuv,wecandenethemulti-levelquadraticordering(MQO)polytopePMQO:=conv(1y1y&#xw-27;:y2f�1;1g;ysatises(2)):Nowthenon-convexequationY=yy&#xw-27;canberelaxedtotheconstraintY�yyT0,whichisconvexduetotheSchur-complementlemma.Moreover,themaindiagonalentriesofYcorrespondtoy2uv,andhencediag(Y)=e,thevectorofallones.Tosimplifyournotation,weintroduceZ=Z(y;Y):=1yTyY;(3)where:=dim(Z)=1+Ppi=1�jVij2
andZ=(zij).WehaveY�yyT0,Z0.Hence,PMQOiscontainedintheelliptopeE:=fZ:diag(Z)=e;Z0g:InordertoexpressconstraintsonyintermsofY,wereformulatethemasquadraticconditionsiny.For(2)thisgivesyuvyvw�yuvyuw�yuwyvw=�1;8u;v;w2Vi;1ip;u__:(4)WecanassignasemidenitecostmatrixCtogived(E)foranygivenfeasibleorderingyandcancomputeMLVObysolvingd=minfhC;Zi:Z2IMQOg,whereIMQO:=fZ:Zpartitionedasin(3),satises(4);Z2E;y2f�1;1gg:Bydroppingtheintegralityofy,wegetabasicsemideniterelaxationforMLVOthatcanbetightenedinmultipleways,e.g.viametric-andLovasz-Schreivercuts.See[4]fordetails.TomaketheSDPcomputationallytractable,weonlymaintainthecon-straintsZ0anddiag(Z)=eexplicitely,anddealwiththeotherconstraintsviaLagrangiandualityusingsubgradientoptimizationtechniques(inparticular,thebundlemethod[15,9]).Weobtainupperboundsviathehyperplaneroundingmethod[12],supplementedbyarepairstrategy.Again,wereferto[4]fordetails.Therein,itisalsoshownthatthisapproachclearlydominates|boththeoreti-callyandpractically|otherapproachesbasedonlinearorquadraticprograms.21