15] InvitedReview Ani ntroducti ontoti metabl i ng D. d e WE RR/ ISwissFederalh~stituteof Technology.Department ofMatllematics,CH-IOIS,Lausanne,Switzerland Abstract. Ahugevari et yoft i met abl i ngmodel s havebeendescri bedintheORliterature;t hey rangefromtheweekl yt i met abl eof aschool t othe schedulingof cour sesor examsinauniversity.Gr aphs andnet wor kshaveproventobeuseful in theformul at i onandsolutionof suchprobl ems.Vari ousmodel swiltbedescri bedwithane. nphasi s ongraphtheoreticalmodels.Keywords: Ti met abl i ng, educat i onnet wor ks,graphs, heuristic,,: 1,Intro~tuetion Duri ngthelastt went yyearsmanycont ri bu- tionsrelatedtot i met abl i nghaveappear edandit willp r o b a N/ c ondnue withthesamer a' efor years.Onereasonforthismaybethehugevariety of pr obl emswhi chareincludedint hefieldof timetabling;anot her reasonisthefactthateduca-tionalmet hodsarechanging,sothatthemodel s havetobemodi fi ed. Fina!ly.~,._,mputiagfacilities arenowavai l abl einmost schoolsand, zs,;cons e quence, theappr oachtot i membl i nghast otake thisphenomenoni nt oaccount. Thismeansinpar- ticularthati nt eract i vemet hodsarenowbecomi ng mor ei mport ant and, furthermore, t heyhavetobe adapt edtomicrocom[~'uters C.,,uailyoneconsi dersdne;.ic,eut bfi agproce.as, t~t:~consisl.~c,f2distinctphases[71: a)First.thecurri cul aaredefi nedforeachclass ReccivegAug=5:~.')S;t N.;,r',h- H,',~!~,~cl Eu : o p e ~ Journ?.!ofOperat : onal R-zsea~c0 19..i9,35)!51-162 or foreachgroupof ~.tudentsandoneha.;to assignthevari ousresources(inmanpower or ,n equi pment ) totheclasses. b) Second, whenanagr eement hasbeen, e- achedconcerni ngtheseassi gnment sof resours, t henonetriestoseewhet her aworkab!edet a-d t i met abl ecanbewor kedout whichiscompat i le wi t hallt heprevi ousl ydef i nedrequirements.(3eneral / ytheroleof comput er s ist ohandl ethe secondphaseandweshallconcent r at eonthist ype of pr obl em. However, oneshoui dment i ont,at someat t ent i onhasal sobeengivent ot hefirst phase, bot hfromat heoret i cal andfromacomou-t at i onal poi nt ofview[15,20]. Inthissurveywecannot descri beallspecial t ypesof t i met abl eswhichhavebeenment i onedup tonow. W,=shallrefert hereadertotheannot at ed bibliograp!~y[29]whi chcoversthecont r i but i ons wbhchappear edbef or e1980andtogeneralsurve,js [5,7,18]. Our pui ~osewillbet,.~ i nt roducethereadertc somebasi cproblem:,whichoccurir~rues,',o.~ the realcases;weshallfirstdescri bethecl a~- t c~c) ; c.~ pr obl emanditsvanatic.ns."thenwe,,-All d~,,cus; thecourseschedulingrr.odeltoge~',erwith.,,o~e closelyrelatedprc:blems.For bot htyp.~softimeta- bl i ngprobl ems, "reshallmai nl ycor c~nt r at eon model sbasedongr aph; andnetwo:lcs.Asob-servedbyMul vey{24], suchmodel sareiateresting.sincenet workpr obl emsareefficientlysolvedeven whentheirsizeislarge;al t hough, generally,reai t i met ab!i ngprobl emscannot beformui at edwith warenet wor kmo:!els,inthema n- ma c z i ne iter~- t Vesoi uf i on,-,,,l., . - ,,r~netwc,,k'-~''~["~I c.ccura!'.?rio'.:s5~ag..?s.O~t~.=v,:'.,hertZ'oretc:.::e ::d,,a~_-'.tag,~:~Ith~fact:!'.z:~ :.[..ey arcr,.!aiveiyeasy t:)handl e[odevisefastheuristicm~,thradsfor Anot her argument l ot j,_,.~tifyiage e r ~nt~.re~:t i~: =-,od~ls.%~:i'npe~,timetab~.i:~gprob2cmsi~tne~c.'. ; an: heuristic: net hcdsf f cr ' .,-anJ!i~:ea~"..irc.e'z.-. i-.linor~,roble,'rc~) areof t end'~ri,c ,~a~dadavtec2 0377-22!7/ 85/ $3. 30' b1985,E!sz'AerSci,.'nzePub2:,he:~p.V.(Nor~.h-[-ioiiand) 152D.deWerm /Ani n t ~ i o n to ametabling fromexactn~t hc, l s dew:lopedfort hesimplecases [30]. Inthistextweshallassumethatt hereaderhas someveryel ement aryknowledge of graph-theore' ;- icalconcepts. Alltermsrelatedt ographsand net workswhicharenot definedherecanbefot md in[t,111. 2.The-dass-teacher model . . ~ . As i mpl e mo a e lWefirstdescri bethebasiccl ass- t eacher model wi t hout includingalIconstraintswhichareusually presentinrealca.,es. Aclasswi l l camsistof asetof s~udentswho followexactlythesameprogramLet C={c 1. . . . . c,~}beasetofcl os es andT,--(t~ . . . . . G} asetofteachers.Wearegi,,enanmxnrequire- ment matrixR=(r~j)where%isthenumber oflecturesinvolvingclassqan, tteacherty. Weshallassumetkatalll ~t ur es have*.he same durat i on(say oneperiod).Gi wmasetofpperiods, theprobel m~stoassigneach!ecturet osome peric~'linsuchawaythatnoteacher(resp.no class)isinvolvedinmorethanonelectureata timeMoreprecisely,ifwedefinex,~ ktobe1if cl l ssc,andteachertjmeetatperi odkand0 olhelwise, wehavetosolve ~aroblemCTI :,.~V" x,i~=r,j( i = 1 . . . . . n . ; j = l , . . , n ) , ( 1):1 ~_,x, j ~~1( i - - 1 . . . . . m ; k =1 . . . . .p ) , (2) i ~ 1 E x , : < l ( : = 1 . . . . .n ; k = ! . . . . .p) , (3) .x~:~ =0or1(i =, 1 . . . . . m ; j =1 . . . . , n;(4) k = 1. . . ., p) ,Withthisformul at i onwemayassociateabi par- titemultigraphG=( C, T . . ~ ) : It.nodesarethe classesandtheteachers;nodec,andno.tetjare linkedbvr,,para!!,:!edges.Ifeachperi adcorre- spot~.ds :oacolor. ~heprobl emconsistsinfinding anassignmentcf or~e amongpcol orst oe,:tch edge ofGinsuchawa-,: thatnot woadjaceF,tedges havethesamecolor;sox,i ~. wiltbe1ifSO'heedge. [ c . t : ] getscolc.rk. Proposi t i on2.1.The r e e x : s t s asolutiont oCT 1 i f fm ~.,r, j