On a Clique-Based Integer Programming Formulation of Vertex Colouring with Applications in Course Timetabling

Ann Op er Res man uscript No. (will be inserted b y the editor) A Sup erno d al F orm ulation of V ertex Colouring w ith Applications in Course Timetabling Edm und K. Burke 1 , Jakub Mare ˇ cek 1 2 ⋆ , Andrew J. P ark es 1 , Hana Rudo v´ a 2 1 Automated Sc heduling, Optimisatio n and Planning Group The Universit y of Nottingham School of Computer Science Nottingham NG8 1BB, UK 2 Masaryk Universit y F a culty of Informatics Botanic k´ a 68a, Brno 602 00, The Czech Republic Received: December 5, 2007 Reviewe d: July 1, 2008 Revised versi on: Oct ob er 22, 2018 Abstract F or man y problems in Sc heduling and Timetabling the choice of an ma thematical pr ogramming for m ulation is determined b y the fo rm u- lation of t he gra ph colouring component. This paper briefly surv eys seven known int ege r progra mming for m ulations of v er tex colouring and in tro duces a new formulation using “sup erno de s ”. In the definition of Geor ge and McIn- t yr e [SIAM J. Numer. Anal. 15 (1978 ), no. 1, 90 – 112], “sup erno de” is a complete subg raph, where ea c h tw o v ertices hav e the same neighbourho o d outside o f the subgr aph. Seen another wa y , the algorithm fo r o btaining the bes t p ossible partition of an arbitrar y g raph into sup erno des, which we give and sho w to b e p olynomial-time, mak es it p ossible to use an y f or m ulation of v ertex multicolouring to encode v ertex colouring. The pow er of this ap- proach is shown on the benchmark pr oblem of Udine Course Timetabling. Results from empir ical tests on DIMACS co louring instances, in addition to ins tances from other timetabling applications, are also provided and dis- cussed. Key w ords vertex colouring, graph colouring, multicolouring, sup erno de, int ege r prog ramming ⋆ Con tact e-mail: jakub@mare cek.cz 2 Edmund K. Burke et al. 1 Introduction Graph colouring (“pro per vertex colo uring”) is a well-known N P - Complete problem (Kar p, 1 972; Garey & Johnson, 197 6). It can b e for m ulated as follows: Given a s imple undire cted, but not necessa r ily connected g raph G = ( V , E ) and a n integer k , decide if it is possible to assign k colours to vertices v ∈ V such that no t wo adjacent vertices { u, v } ∈ E are assigned the same colour . Graph colouring has a num ber of applications, ra nging from universit y timetabling (Car ter & Lap orte, 1997; Schaerf, 19 99; Petrovic & Burke, 2 0 04) and frequency assignment in cellular netw orks (Aar dal, Ho esel, Koster, & Mannino, 20 07), to registry allocatio n in compilers ( Springer & Thomas, 19 94) a nd a utomating differentiation (Gebremedhin, Manne, & Pothen, 2005 ). Graph colour ing is a challenging problem: As well as b eing N P -hard to solve e x actly , the minim um n um b er of co lours needed to co lour a g raph is als o N P -Hard to approximate within a factor of | V | 1 − ǫ for a n y ǫ > 0, unless N P = P ( Kra j ´ ıˇ cek, 199 7 ; F eige & Kilian, 1998; Zuck erman, 2007). Also, there ar e still dens e rando m ins tances on 125 vertices from the Seco nd DIMA CS Implem entation Challenge announced in 1 992 (Jo hnson & T r ic k, 1996), for which the decision pr o blem cannot be solved within reasona ble time limits (M´ endez- D ´ ıaz & Zabala , 2008 ), How ever, it is often p ossible to solve considerably larg e r instances in practice, by exploiting applicatio n- sp ecific structure of the graphs. Spring er a nd Thomas (1994) ha ve, for in- stance, shown that graph colouring in sp ecial cases o f register allo cation in compilers is po lynomially so lv able. In cas es that a re not p o lynomially solv able, exact s olv ers intro duced in the past t wen ty years ha ve predominan tly been bas ed on a bra nch and bo und/cut pr o cedure with linear pro gramming rela xations. There a re a wide v ariet y of such integer linea r pr o gramming appro ac hes to mo delling g raph colouring. A num b er of author s, including Z abala a nd M´ endez-D ´ ıaz (200 2; 2006; 2008), hav e used a natur al assig nmen t-type fo r m ulation. Williams and Y an (20 01) hav e s tudied a for m ulation with prece de nc e constr a in ts. Lee (2002) and Lee and Marg ot (20 07) have studied a binary enco ded formula- tion. Mehrotra and T r ic k (1996 ) and more recently (Sc hindl, 2 0 04; Hansen, Labb´ e, & Schindl, 20 05) hav e b een using formulations ba sed on indep enden t sets. Barb osa et al. (200 4) have b een experimenting with encoding s bas e d on ac y clic orientations. Finally , the most rece nt formulation by Campˆ elo , Camp os, and Co rr ˆ ea (200 8) is based on asymmetric repr esen tatives. These seven enco dings o f gra ph colouring, often toge ther with the corresp onding int ege r prog ramming formulations, ar e survey ed in Section 2. In Section 3, we fir st rev iew the concept o f a “sup erno de”, a complete subset of vertices o f a gra ph, where e a c h t wo vertices hav e the same neig hbo urhoo ds outside of the subset; this concept has b een descr ibed many times previously (Georg e & McIn tyre, 1978; Duff & Reid, 1983; Eisenstat, Elman, Sc hultz, & Sher- man, 1984). See Figur e 1 for a simple illustration. Next, we show that the partition o f a gr aph in to super nodes, obtainable in po ly nomial time, pro - Sup ernod al F orm ulation of Graph Colouring 3 Fig. 1: Example of a graph and a partition of its vertex-set into sup ernod es. Notice sup ernodes B ′ and C ′ need to b e assigned tw o distinct colours each, distinct from th e colour(s) assigned to A ′ and D ′ . Within eac h sup erno de, colours can b e interc hanged freely . F or a more complex example, see Figure 5. A B 2 B 1 C 2 C 1 D A ′ B ′ C ′ D ′ T able 1: Integer programming form ulations of graph colouring: Based on V ariables Constrain ts Selected references V ertices k | V | | V | + k | E | M ´ en dez-D ´ ıaz and Zabala (Standard) (2002, 2006, 2008) Binary Enco d ing ⌈ log 2 k ⌉ | V | Exp. many Lee (2002) Max. Indep endent S ets Exp. m any | V | + 1 Mehrotra and T rick (1996) Any Indep endent S ets Exp. m any | V | + 1 Hansen et al. (2005) Precedencies O ( | V | 2 ) | E | Williams and Y an (2001) Acyclic Orientations | E | Exp. m any Barbosa et al. (2004) Asymmetric Represent. O ( | E | ) O ( | V | | E | ) Camp ˆ elo et al. (2008) Sup ernod es k | Q | | Q | + k | E ′ | This pap er vides a tra ns formation of graph colouring t o graph m ulticolouring. Hence, we ca n use the standard bina ry integer formulation of multicolouring, with binary decision v ar iable x ij is set to one, if any mem ber of sup erno de i is assigned colour j , fo r graph co lo uring. This tr anslates to new for m ulations for n umerous problems in Scheduling and Timetabling. An illustrativ e ex- ample of for m ulations of Udine Course Timeta bling (Gasp ero & Schaerf, 2003, 2006) is g iv en in Section 4 . The pa per is concluded with a discussion of the empirical tests we ca rried o ut in Section 5. 2 Known F orm ulations of Graph Colouring In g raph colouring, w e assume we ar e giv en a simple undirected, but not necessarily co nnected graph G = ( V , E ) a nd an integer k . Integer pr ogram- ming form ulatio ns of the decisio n version of the graph c olouring problem hav e feas ible integer solutio ns if and only if it is poss ible to assign colo urs K = { 1 , . . . , k } to v ertices v ∈ V of G such that no t wo adjacen t vertices { u, v } ∈ E are assigned the same c o lour. Although the minim um v a lue of k is gener ally ha r d to approximate, it is of cour se alwa ys p ossible to pick k = | V | , a nd for real- life gr aphs, heuristics based on lo cal search with suitable pre-pro cessing often p erform well (Galinier & Hertz, 200 6). Estima- tors of the minimal k a r e a lso av a ila ble for some classes of random graphs 4 Edmund K. Burke et al. (Ac hlioptas & Naor, 200 5). Notice that the decision version of the problem with fixed k , rather than the optimisation version lo oking fo r minimal k , is used in many applications. F o r instance in sc ho ol timetabling (Schaerf, 1999), k is usually fixed to the num ber of pe r iods p er week. Although there are at lea st seven p ossible enco dings of feas ible solu- tions a nd hence seven different integer pr ogramming formulations of gra ph colouring, as far as we a re aware, there is no survey ar ticle or empirical co m- parison av aila ble in the litera ture. M´ endez-D ´ ıaz and Zabala (2008) compare four classes of cuts using the standard formulation and Prestwich (2003) compares five encodings of gr a ph colouring in to pro positiona l satisfiability testing. This section elab orates on the br ief ov erview provided in T a ble 1. Unless stated otherwise, w e consider the decision version of the prob- lem. In some cases, co nstrain ts necessary to reaching optimality are also men tioned. Notice, ho wev er, there hav e often b een described many cla sses of a dditio na l constra ints, which can b e a dded dynamically in a branch and cut pro cedure. 2.1 The S tandar d F ormulation The natural assignmen t-type form ulation of graph co lo uring uses k | V | bi- nary v ariables: x v, c = ( 1 if vertex v is colo ur ed with co lour c 0 otherwise (1) sub ject to k | E | c o nstraint s: k X c =1 x v, c = 1 ∀ vertices v ∈ V (2) x u,c + x v, c ≤ 1 ∀ colo urs c ∈ K ∀ edges { u, v } ∈ E (3) This for mulation alone produces pro v ably p oo r linear pr ogramming re- laxations (Caprara, 19 98). Mehrotr a and T ric k (1 996) give the example of x v, c = 1 /k fo r all vertices v ∈ V and for a ll colours c , which is feasible when k ≥ 2. Howev er, a n um b er of cla sses of s tr ong v alid ineq ua lities have bee n des cribed for this for this formulation, most notably by Zabala and M´ endez-D ´ ıaz (200 2; 2006; 20 0 8), and (Campˆ elo , Corrˆ ea, & F r ota, 200 3 ), either supplanting or replacing per- e dge constraints (3) . Branch-and-cut co des using suitable implementations of separation routines have pro duced a num b e r of optimal v alues and present-best b ounds for the b enchmark established by Johnso n and T rick (19 96) (Zabala & M´ endez-D ´ ıaz, 200 6 ). Sup ernod al F orm ulation of Graph Colouring 5 2.2 Extension: Synch r onisation with Gener al Inte ger V ariables Williams and Y an (200 1) have noted that the s ta ndard form ulation could be extended with | V | additional gener al integer v ariables X , w her e X v = c if colour c is us ed to co lour vertex v , s ub ject to | V | additional constr ain ts: k X c =1 cx v, c = X v ∀ vertices v ∈ V (4) This extens io n can be a pplied together with custom branching r ule s with some success in some timetabling problems where, fo r instance, lectures should b e timetabled b efore lab oratory se s sions. 2.3 The I n dep endent Set F ormulation One of the fir s t a lternativ e formulations w as pr o posed by Mehrotra and T rick (1996). It is ba s ed o n set I o f maximal independent sets. (Subset S ⊆ V o f gra ph G = ( V , E ) is defined to b e indep endent, if no tw o u, v ∈ S form an edge { u , v } ∈ E .) There are an expo nen tial num b er of binary v ariables: x i = ( 1 if independent s et i is as s igned a sing le colour 0 otherwise (5) sub ject to | V | + 1 constraints: X i ∈ I x i ≤ k (6) X i ∈ I , s.t. v ∈ i x i ≥ 1 ∀ vertices v ∈ V (7) F or pr ocess ing any but the sma llest of instances , such a formulation ob- viously requires very go o d routines for finding ma ximal independent sets and for adding them to the linear programming subpr oblems o n-the-fly b y the means of column g eneration. It should also be noted tha t solutio ns ob- tained using this form ulation requir e a certain amount o f p ost-pro cessing , if co nstraint s (7) remain inequalities. Alterna tiv ely , the pro blem could b e reformulated so that I comprises a ll independent sets, not only maximal independent sets. In the p er-vertex constraints (7) , inequa lit y can then b e replaced with equality (Mehro tra & T rick, 1996). The orig ina l implementa- tion of Mehrotra and T rick pr oduced exceptionally goo d re sults (Mehrotra & T rick, 199 6), but later reimplementation of Schindl (2004) and Hansen et al. (2005) failed to match the exceptional p erformance. It se e ms also rather difficult to ada pt this for m ulation to extensions of vertex colouring such as the Udine Course Timetabling, which will be in tro duced in Sectio n 4. 6 Edmund K. Burke et al. 2.4 The S che duling F ormulation (with Pr e c e denc e Constr aints) Many resea rc her s from a co nstraint pr ogramming background deal with graph co louring in ter ms of multiple s im ultaneously applied all differ ent constraints. In an assignment A : V → D of v a lues from a finite domain D to v ariables V , applying the al l differ ent constra in t on a subset W ⊂ V stipulates that ther e hav e to b e | W | distinct v alues a ssigned to elements of W . Setting all differ ent ( V ) then makes assignment A in- jective. The case of a single al l diff erent cons tr ain t is easy to s olv e, as it represents bipar tite matc hing. The case o f tw o s im ultaneously ap- plied all differ ent constra in ts was studied by Appa, Mago s , and Mourtos (2005). The gener al case of multiple simultaneously applied all di fferent constraints is, in some sense, equiv a len t to graph colo uring. If we take, for example, the set of v ariables X defined in Section 2.2 , co ns train ts (3) im- plement | E | all differ ent constra in ts to pairs of elements of X . Williams and Y an (2001) ha ve compa red this standard integer progr amming fo rm u- lation o f the all differ ent constra in t (of Section 2.1) with a formulation using pr ecedence constra in ts. Their w ork leads to a formulation of vertex colouring us ing | V | integer v ariables, where X v = c if colour c is used to colour vertex v , and 1 2 | V | ( | V | − 1) additional binar y v ariables x u,v , defined for u < v : x u,v = ( 1 if for vertices u , v holds X u < X v 0 otherwise (8) sub ject to | E | precedence constra in ts: x u,v + x v, u = 1 ∀ edges { u, v } ∈ E (9) (10) How ever, in the exp e rience o f b oth Williams a nd Y an (200 1) and the authors, this fo rm ulation do es not o ffer particular ly s trong relaxa tions. T obias Ach terb erg (p ersonal comm unicatio n) suggested using another enco ding inspired by scheduling: x u,m = ( 1 if vertex v is coloured by c ≤ m 0 otherwise (11) This enco ding is, as far as we know, also untested. 2.5 The Binary Enc o de d F ormulation In his studies of the all differ ent p olyhedron, Lee (2002) and Lee and Margot (200 7 ) have intro duce d a formulation of binary enco ding using ⌈ log 2 k ⌉ | V | binary v ariables: Sup ernod al F orm ulation of Graph Colouring 7 Fig. 2: Tw o encod in gs of a particular colouring of th e graph from Figure 5: Indep endent set Used? { Math 1 } 0 { Math 2 } 1 { Math 3 } 1 { Math 4 } 1 { Algo 1 } 1 { Algo 2 } 1 { Algo 3 } 1 { Phy } 0 { Math 1 , Phy } 1 { Math 2 , Phy } 0 { Math 3 , Phy } 0 { Math 4 , Phy } 0 (a) An Enco ding Using In d e- p endent Sets V ertex Colo ur Bit 1 Bit 2 Bit 3 Math 1 1 0 0 Math 2 0 1 0 Math 3 1 1 0 Math 4 0 0 1 Algo 1 1 0 1 Algo 2 0 1 1 Algo 3 1 1 1 Phy 1 0 0 (b) The Binary Enco ding x v, b = ( 1 if vertex v is assig ned co lour having bit b set to 1 0 otherwise (12) Lee and Margot (200 7 ) als o describ ed three br oad clas s es o f a pplicable inequalities (“genera l blo ck inequalities” , “ matc hing inequalities”, “switched walk inequalities”), each exp onen tially lar ge in | V | . W e conjecture, but can- not prov e, these include all inequalities introduced b y Za ba la and M´ endez- D ´ ıa z (2006 ), when pro jected to th e a ppropriate space. How ever, the devel- opment of sepa ration ro utines for suc h general inequalities is by no means straightforward (Lee & Margot, 2007). In the context of edge colouring o f graphs, it o nly r emains to decide if a g raph req uir es more colours tha n the maximum deg ree of vertices in the gr aph. The c omputationally exp en- sive separa tion o f gener al blo c k inequalities could th us p erhaps b e offset by having to elimina te substa n tially fewer v aria bles in the branch-and-cut pr o- cedure (Lee & Marg ot, 2007). In theory , such a n argument could p erhaps also a pply to co louring o f dense r a ndom graphs (Bollob´ as, 2001 ), where the chromatic num b e r was s ho wn to b e almost surely one of tw o known v alues (Ac hlioptas & Naor, 2005). How ever, exp erimental results do not seem to be conclusive; not even in the case o f edge co louring (Lee & Mar got, 2007 ). 2.6 Enc o di ng Using A cyclic Orientations In the context of exp e rimen tal formulations o f gr aph co louring, we also men- tion acyclic orie ntations, an enco ding ba s ed the Ga llai-Roy-Vita ver theore m (Gallai, 19 6 8; Roy, 1967 ; Vitav er, 1 9 62): dire cted gr aphs, which contain no directed simple pa th o f length ≥ k , k ≥ 1, are k -co lorable. An acyclic ori- ent atio n G ′ = ( V , E ′ ) of an undirected G = ( V , E ) is then o b viously a 8 Edmund K. Burke et al. directed gr aph such that for each { u, v } ∈ E , there is either ( u, v ) ∈ E ′ or ( v , u ) ∈ E ′ , and there is no directed cycle in G ′ . F or further references, see also W erra and Hansen (2 003) and Ne ˇ set ˇ ril a nd T ardif (2008). T ogether with an algorithm enum er ating all p ossible acyclic orientations (Bar bosa & Szwarcfiter, 1999), this could provide a basis for a column genera tion a lgo- rithm for g raph co louring. There are s ome exp erimen ts with metaheuristics using this enc o ding (Barbos a et a l., 20 04). The only implementation using the linear pr o gramming rela xations with this enco ding the a uthors a r e aw ar e of, how ever, is a n unpublished work of Rosa Mar ia Videira de Figueiredo. 2.7 F ormulation Using Asymmetric R epr esen t atives Finally , the most recently published alternative formulation o f gra ph colour- ing is b y Camp ˆ elo et al. (2 008), although it do es s tem fr om their previous studies of gra ph colour ing (Campˆ elo et al., 2003). There are | V | + | V | 2 − | E | binary v ariables x u,v , where x u,v is defined for u , v ∈ V , u 6 = v , and { u, v } / ∈ E : x u,v = ( 1 if vertices u , v share o ne colour and u represents v 0 otherwise (13) Each indep endent set, which is assig ned a unique colour, is as signed a unique vertex (“representative”) r epresen ting the independent set. This can b e done using a num ber of constra ints cubic in | V | . Campˆ elo et al. (2008) then establish an or der on the vertex set V , which induces an acyclic orientation int ro duced in Section 2.6. This enables addition o f a num b er of symmetry-brea king constraints. No empirical results are av ailable, thoug h, as Campˆ elo et al. (2008) rep ortedly hav e problems designing separa tion routines for the cutting planes they prop ose. 3 The Mai n Result In this section, we prop ose ano ther formulation, ba s ed on a particular type of clique partition. Let us reiter ate, how ever, the definition of a cliq ue partition first: Definition 1 The cliq ue partition of gr aph G = ( V , E ) is a p artition Q of vertic es V , such t hat fo r al l sets q ∈ Q , al l v ∈ q ar e p airwise adjac ent in G . Notice we use v ∈ q ∈ Q only to denote that vertex v in the o riginal vertex-set V is an element of a clique represe n ted by q in the clique pa rtition Q . Hence, ther e is no need to interpret this a s the use of hyp e r-graphs. As is w ell known, the pro blem o f finding the minimum cardinality of a clique par tition, ¯ χ ( G ), is N P - Hard in general gra phs and as hard to Sup ernod al F orm ulation of Graph Colouring 9 Fig. 3: Tw o more enco dings of a p articular colouring of the graph from Figure 5. Identical row headings are not rep eated twice. V 1 V ertex V 2 Math 1 Math 2 Math 3 Math 4 Algo 1 Algo 2 Algo 3 Phy Math 1 1 1 1 1 1 1 0 Math 2 0 1 1 1 1 1 0 Math 3 0 0 1 1 1 1 0 Math 4 0 0 0 1 1 1 0 Algo 1 0 0 0 0 1 1 0 Algo 2 0 0 0 0 0 1 0 Algo 3 0 0 0 0 0 0 0 Phy 0 1 1 1 1 1 1 (a) The Scheduling Enco ding V ertex V 2 Math 1 Math 2 Math 3 Math 4 Algo 1 Algo 2 Algo 3 Phy 0 1 1 0 1 0 1 0 1 1 1 0 0 0 0 0 (b) The Encod ing Using Asymmetric Representativ es approximate a s gr aph colouring itself (see Minimum-Cliq ue-P ar tition in Crescenzi, Ka nn, Halld´ orsson, Kar pinski, & W o eginger, 200 5 ). Indeed, ¯ χ ( G ) = χ ( ¯ G ), wher e χ ( ¯ G ) is the minimum num ber of co lours needed to colour the complement graph. Another direc tion o f ar riving a t probabilis tic bo unds on ¯ χ ( G ) co uld, p erhaps, follow from proba bilistic r e sults of Molloy and Reed (2002, Chapter 11) for m a x imal cliques. Notice, how ever, we do not req uire minimalit y in the definition, and hence V is the trivia l clique partition of graph G . Next, w e in tro duce the indistinguishability r elation b etw een v ertices of a g raph: Definition 2 Two vertic es u, v ∈ V of a gr aph G = ( V , E ) ar e indistin- guishable , if and only if they ar e adjac ent and have identic al close d neigh- b ourho o ds; that is: { w | { u , w } ∈ E } ∪ { u } is the same as { w | { v, w } ∈ E } ∪ { v } . This rela tion has b een studied previously in the context o f pivoting in ma tr ix factor isation, in connec tio n with mass elimination (Geo rge & 10 Edmund K. Burke et al. McInt yre, 19 78), sup erv ariables (Duff & Reid, 1983 ), and prototype ver- tices (Eise nstat et al., 1 984). It is ea s y to observe the indistinguishability relation is reflexive, symmetric, and transitive. Hence: Lemma 1 The indistinguishabi lity r elatio n is an e quivalenc e. Next, we define the particular type of c liq ue partition we are interested in: Definition 3 The r e versible clique partition Q of a gr aph G = ( V , E ) is the clique p artition of minimum c ar dinality such that e ach sup erno de q ∈ Q r epr esents a class of e quival enc e in a indistinguisha bility r elation on G . This means that for eac h superno de q ∈ Q of the reversible clique par - tition ( Q, E ′ ), each tw o vertices u, v ∈ q are indis ting uishable. As usual, we will b e in terested a lso in the graph induced by the clique par tition: Definition 4 The g raph induced by reversible clique partition Q of gr aph G = ( V , E ) is the gr aph G ′ = ( Q, E ′ ) , wher e E ′ = {{ q u , q v }|{ u, v } ∈ E , q u , q v ∈ Q, q u 6 = q v , u ∈ q u , v ∈ q v , } . The use o f the word induc e d in this context is r easonable, b ecause it co r- resp onds to a subgr aph induced by taking a subset of the o riginal vertex set with a single (ar bitrary) repres en tative o f each sup erno de. T he “rev ers ibil- it y” of the clique partition is, indeed, rather a str ict requir emen t, which enables us to formulate the following: Definition 5 Algo r ithm A Input: Gr aph G = ( V , E ) Output: R eversi ble clique p artition Q of G 1. Constru ct an aux ilia ry gr aph H = ( V , F ) , wher e ther e is an e dge { u, v } ∈ F , if and only if ther e is an e dge { u, v } ∈ E and vertic es u and v ar e indistinguishable in G 2. Run depth-first se ar ch on H to obtai n c ol le ction Q of c onne cte d c omp o- nents of H 3. Return Q W e can easily deduce th at: Lemma 2 Algorithm A pr o duc es a r eversible cli que p artition. F rom Step 1 , it is clear eac h elemen t of the collection we r eturn corre- sp onds to a class of equiv alence in the indistinguishability relation on G . By transitivity of the indistinguisha bility relation, it is clear the algorithm pro duces a clique par titio n. Now imagine there is a sma ller clique partition R cor respo nds t he indistinguishability relation on G . It is ea s y to see the contradiction. Hence, the algo rithm o btains a reversible clique partition. F urthermore: Sup ernod al F orm ulation of Graph Colouring 11 Lemma 3 Algorithm A runs in time O ( | V | | E | ) . Given Algo r ithm A , we c an straigh tforwardly reform ulate the problem of vertex colouring as the problem of m ulticolour ing of the corr esponding reversible clique partition, where by multicolouring, we mean: Definition 6 The pr oble m of multicolouring of a gr aph G = ( V , E ) with a finite set of c olours K = { 1 , . . . , k } , which is given to gether with demand function f : V → N , is to obtain is a map ping c : V → 2 K , s u ch that for al l v ∈ V : | c ( v ) | = f ( v ) and for al l { u, v } ∈ E , c ( u ) ∩ c ( v ) = ∅ . It makes sense to r e quir e S v ∈ V c ( v ) = K . Notice that multicolouring with s ets of uniform cardina lit y has b een studied under the name of set colouring, for exa mple by Stahl (1976), Bollobas and Thomason (197 9), and more recen tly used also by Duran et al. (2002; 200 6). Other v ariants of the pro blems a re surveyed by Halld´ orsson and Kortsa rz (2 004) and Aardal et al. (2007 ). Mehrotra and T rick (2007) seem to ha ve the pres en t-b est solver for multicolouring. F rom Lemma 2, it is ea sy to observe that Algorithm A provides a trans- formation of vertex c olouring to vertex m ulticolour ing. Hence, the s tan- dard for m ulation of vertex mult ico louring c a n a lso b e us e d as a for m ulation of v er tex colour ing. Given the graph G ′ = ( Q, E ′ ) induced b y reversible clique partition Q of g raph G = ( V , E ) tog ether with the dema nd function f : V → N , spec ifying the num b er f ( q ) o f colour s to attac h to eac h v ertex q ∈ Q out o f the se t K = { 1 , . . . , k } , we can use an int eg er programming formulation with k | Q | binary v ariables: x q,c = ( 1 if colo ur c is included in the set assigned to q ∈ Q 0 otherwise (14) sub ject to | Q | + k | E ′ | constraints: k X c =1 x q,c = f ( q ) ∀ q ∈ Q (15) x u ′ ,c + x v ′ ,c ≤ 1 ∀ c ∈ K ∀{ u ′ , v ′ } ∈ E ′ (16) See Figur e 4 for an example. It is easy to see that there exists a prop er vertex colouring of G = ( V , E ) with k colours, if and only if t her e exists a m ulticolo uring o f a reversible clique partition ( Q, E ′ ) of G with k colour s, which exists if and only if the integer pro gramming formulation has a feasible solution for the given instance. When a graph has only a trivial r ev ersible clique partition, this for m ulation is reduce d to the standard formulation. It th us r emains N P -Complete to decide , if there exis ts a multicolouring o f G ′ with f ( q ) using k colour s. Nevertheless, the prop osed formulation brea ks some symmetries inher e nt in the standar d vertex co louring for m ulation, 12 Edmund K. Burke et al. Fig. 4: The standard and the prop osed enco ding of a particula r col ouring of t he graph from Figure 5: V ertex Colour 1 2 3 4 5 6 7 Math 1 1 0 0 0 0 0 0 Math 2 0 1 0 0 0 0 0 Math 3 0 0 1 0 0 0 0 Math 4 0 0 0 1 0 0 0 Algo 1 0 0 0 0 1 0 0 Algo 2 0 0 0 0 0 1 0 Algo 3 0 0 0 0 0 0 1 Phy 1 0 0 0 0 0 0 (a) The S tandard Enco ding P artition Colour 1 2 3 4 5 6 7 Math ′ 1 1 1 1 0 0 0 Algo ′ 0 0 0 0 1 1 1 Phy ′ 1 0 0 0 0 0 0 (b) The Prop osed Encod ing which assigns unique colour s (or “lab els”) to individual vertices. If there was a trivia l integer progr amming solver, using neither bo unding, no r cuts, this formulation should reduce its sea rc h spac e and r un time by the factor of: Y q ∈ Q | q | ! when compared to the standard form ulation of Section 2.1 . Although it is m uch mor e difficult to predict r un times in mode r n integer pro gramming solvers, it is obvious that there a re k ( | V | − | Q | ) fewer v ariables, in the prop osed formulation than in the standar d one. It seems that the num b er of constraints is also reduced, often by more than k ( | V | − | Q | ), w itho ut making the constr ain t matrix consider ably denser. Hence, reduction in r un time o f the o rder of | Q | / | V | could p erhaps b e expected. F or empirical results, see Section 5. 4 An Appl i cation in Course Timetabling In general, a comparis o n of formulations of graph colouring is non-trivial. Both enco dings ba s ed on indep enden t sets and r epresent atives introduce less symmetry 1 than the sta nda rd formulation intro duced in Sec tio n 2.1 or 1 When we address the question of reducing or breaking symmetry b elow , the statements hold, when symmetry is thought of as the number of solutions of the instance of integ er programming with th e b est p ossible cost, corresp onding to, in some sense, Sup ernod al F orm ulation of Graph Colouring 13 binary enco ding. Although they neatly partition the set of vertices, witho ut assigning unique la bels to individual pa rtitions, their merits are har d to quantify , as any empirical results are dep enden t on a par ticula r implemen- tation of separ ation a nd pricing routines, whic h ha ve not been extensively studied th us far . Another imp ortant asp ect is extensibilit y of the v arious formulations of gra ph colouring. Many real-world applicatio ns necessitate formulation of complex measures of the qualit y o f feasible solutions (“key per formance indicator s”), w hich seem to b e hard to formulate using an ex- po nen tial num b er o f v a riables representing independent sets ( Mehrotra & T rick, 1996; Hansen e t al., 200 5 ) or using the bina ry e ncoding of Lee (20 02). One such applicatio n arises in a num b er of universities ( Burk e, W er ra, & Kingston, 2004 ): co urse timetabling. In educational timetabling, considera ble r esources ca n b e wasted by low utilisation of teaching s pace ( Beyrouthy et al., 2008 ). Sp ecific timetabling problems v ary widely from instit ution to institu tion. Most problems, how- ever, share a c o mmon mo del: – set E of even ts is given, together with a subset o f its p ow erset A , where for all distinct “enr olmen ts” (or “co nflict gr o ups” or “curr icula’ ’) a ∈ A , even ts e ∈ a cannot take place at the same time – assignment o f event s to | P | time pe r iods is desired, such that a ll distinct enrolments ar e ho no ured a nd there a re a t most | R | even ts tak ing pla c e at one per iod, where | P | is the num be r of perio ds p er week a nd | R | is the nu mber of av aila ble ro oms. This mo del is, indeed, a s traight for w ard applica tion of | R | -bo unded | P | - colouring. In the graph to be coloured (the “ conflict gra ph” ), vertices r ep- resent even ts, t w o v ertices are adjacen t if the corresp onding even ts are in- cluded in a single enr olmen t, and assignmen t of perio ds to ev ents is repre- sented by as signmen t of | P | colour s to | E | vertices, such that adjacent ver- tices are ass igned different colo urs and each c olour is used a t most | R | times. F or an illustrative exa mple, see Fig ur e 5. F or further g raph-theoretical foun- dations, see Handb o ok of Gr aph The ory (Gro ss & Y ellen, 200 4), esp ecially Section 5.6 (Burke et al., 2004). The most r igorous studies o f integer pr o- gramming formulations of this mo del, including comp etitiv e bra nc h-and-cut implemen tations , a re b y Av ella and V asil’ev (2005 ) a nd M ´ endez- D ´ ıaz a nd Zabala (20 08). F or other r ecen t r esearch dir ections, see Burke and Petro- vic (2002). How ever, it s eems obvious that this mo del is rather re moved from the needs o f r e al-life applications, although given the complexity o f vertex colour ing, where the pre sen t-b est so lv ers hav e difficulties with dense instances o n 125 vertices (Zabala & M´ endez-D ´ ıaz, 20 06), it also pre s en ts an int ere sting challenge. a single configuration. The assignmen t of colours is irrelev ant, for example, as long as we are giv en the appropriate vertex-partition. Presumably , the statements could also hold for other defi nitions of symmetry as well . 14 Edmund K. Burke et al. Fig. 5: An example from timetabling. Imagine one student takes Algorithms and Mathematics (with th ree and four lectures p er week), and another one takes Al- gorithms and Physics (with a s ingle lectu re per week); no tw o lectures attend ed by one student can take place at the same t ime. The correspond ing reversi ble clique partition G ′ = ( Q, E ′ ): Q = { Phy ′ , Algo ′ , Math ′ } , E ′ = {{ Math ′ , Algo ′ } , { Algo ′ , Phy ′ }} Phy ′ Algo ′ Math ′ The original conflict graph G = ( V , E ): V = { Phy , Algo 1 , Algo 2 , Algo 3 , Math 1 , Math 2 , Math 3 , Math 4 } E = {{ u, v } | u, v ∈ V , u 6 = v } \ {{ Phy , Math 1 } , { Phy , Math 2 } , { Ph y , Math 3 } , { Phy , Math 4 }} Phy Algo 1 Algo 2 Algo 3 Math 1 Math 2 Math 3 Math 4 In this pap er, we use a mo del of co urse timetabling pro posed by Schaerf and Di Gaspero (2003, 2006) a t the Universit y of Udine. In Udine Course Timetabling, the basic mo del is extended s o that: – ev ents are g roupe d in to disjoint sets, ca lled cour ses, w ith even ts of o ne course ta king place at different times and b eing fr e ely interc hangea ble – only imp ortant distinct enrolments, or non-disjoint sets of cours es pre- scrib ed to v arious groups of students, are identifi ed – capacities of individual cla ssro oms a nd enro lmen ts in individua l course s are also g iven, and a ssignment of ev ents to ro oms as well as p erio ds is desired, minimising v alue of an ob jective function What makes the extension more difficult (by or ders o f mag nitude) than the basic mo del, how ever, is the ob jective function, consis ting of a linear combination of thr ee key p erformance indicators: – the num ber of students left without a sea t at an ev ent, summed a cross all events – the differ e nc e b etw een the prescr ibed minimum num ber o f distinct days of instr uction for a cour s e a nd the a ctual num b er of distinct days, when even ts of the course ar e held, summed across all courses, where the difference is positive Sup ernod al F orm ulation of Graph Colouring 15 – the num ber of events o ccurring o utside of a con tinuous blo c k of tw o or more even ts in a timetable for an impo rtan t dis tinct enro lmen t, summed across a ll imp ortant distinct enro lments Notice that the third key performanc e indicator essentially consis ts o f the sum o f the num ber o f bre aks in individual timetable s of individual students or gr oups o f studen ts, plus the num b er o f s ingle courses on a single day in the timetables . Its modelling proves to b e very difficult (Burke, Mareˇ cek, Park es, & Rudo v´ a, 20 08) and the present b est solv ers y ield “p oo r results” (Av ella & V asil’ev, 2 005). See also Schimmelpfeng and Helb er (2 007) for another exa mple of a timetabling problem with a n umber of s oft-constraints, together with an in tere s ting integer prog ramming for m ulation. In a further extensio n of the basic mo del, not studied in this pap er, one relaxes also the colouring comp onent . V ertices o f an edge-weighted conflict graph then ha ve to b e partitioned in to | P | disjoint subsets such that the sum of weigh ts a ttac hed to edges with b oth end-p oints in a single subset is minimised ( K iaer & Y elle n, 199 2). The weight of an edge { e 1 , e 2 } ∈ E can b e deter mined, for instance, by the n umber of students enrolled in bo th events e 1 and e 2 . Ob vious ly , if the conflict graph is | P | -colo urable, a prop er co lo uring is found. Such a mo del is employ ed, for ins tance, a t P urdue Univ er sit y (Rudov´ a & Murray, 2003 ; Mur ra y , M¨ uller, & Rudov´ a , 2007 ). 4.1 Notation for Course Timetabling In order to present timetabling applications of the prop osed formulation of graph colouring, w e hav e to in tro duce some notation. In the co n text of course timetabling, it is customary to refer to vertices a s even ts and c olours as p erio ds. In addition to a p eriod, each even t is assigned a lso a ro om, a nd there can b e, a t most, a g iv en num b er of even ts ta king place a t ea ch p erio d. Using this con ven tion and the no tation presented in T able 2 , the standard int ege r prog ramming for m ulation of co urse timetabling is written a s: T p,r,e = ( 1 if even t e is taught in ro om r a t p erio d p 0 otherwise (17) 16 Edmund K. Burke et al. T able 2: The notation used in our integer programming form ulation of Udine Course Timetabling. R set of ro oms Capacit y r the subset of p eriods p ertaining to day d P set of p eriods D set of days P erio ds d the subset of p eriods p ertaining to day d C set of courses MinDa ys c the recommended minimum number of days of instruction for course c Students c num b er of stud en ts enrolled in course c E set of even ts E c the subset of even ts p ertaining to course c T set of teachers T eaches t the subset of courses taught by teacher t U set of identifiers of d istinct enrolments HasC u the subset of courses p ertaining to curriculum u X r X p T p,r,e = 1 ∀ event s e ∈ E (18) X e T p,r,e ≤ 1 ∀ p erio ds p ∈ P ∀ ro oms r ∈ R (19) X r X e ∈ c T p,r,e ≤ 1 ∀ p erio ds p ∈ P ∀ courses c ∈ C (20) X r X c ∈ T eaches t X e ∈ c T p,r,e ≤ 1 ∀ p erio ds p ∈ P ∀ teachers t ∈ T (21) X r X c ∈ HasC u X e ∈ c T p,r,e ≤ 1 ∀ p erio ds p ∈ P ∀ curricula u ∈ U (22) This corres ponds to the s tandard formulation of graph colour ing in- tro duced in Section 2.1. Constra in ts (18 ) ens ure each even t is assigned a time-place slot and constraints (19 ) ensure there is at mo st one even t tak- ing place in a g iv en ro om at a p eriod. Finally , the packing-type constraints (20)–(22) s tipulate there should be no conflicts. No tice that constr ain ts (22 ) make constr ain ts (20) redundant, unless ther e ar e cours es no t included in any enrolment. In a simila r spirit, the formulation intro duced in Section 3 can b e written, with cour ses as supe rnodes , as follows: Sup ernod al F orm ulation of Graph Colouring 17 T p,r,c = ( 1 if some even t of course c is taught in r oo m r at p erio d p 0 otherwise (23) X r X p T p,r,c = | E c | ∀ courses c ∈ C (24) X c T p,r,c ≤ 1 ∀ p erio ds p ∈ P ∀ ro oms r ∈ R ( 25 ) X r T p,r,c ≤ 1 ∀ p erio ds p ∈ P ∀ cours e s c ∈ C (26) X r X c ∈ T eaches t T p,r,c ≤ 1 ∀ p erio ds p ∈ P ∀ teachers t ∈ T (27) X r X c ∈ HasC u T p,r,e ≤ 1 ∀ p erio ds p ∈ P ∀ curricula u ∈ U (28) What mak es real-life course timetabling v a s tly more difficult than this formulation o f graph colour ing, are complex measures of the quality of fea- sible timetables, which are b est illus trated by considering an example. 4.2 F ormulation of Udine Co u rs e Timetabling Udine Course Timetabling , in tro duced in Section 4, is an established b enc h- mark in the field o f course timetabling with complex per formance indicators . Out of the three key p erformance indicator s in Udine Course Timetabling, the minimisa tion of the n umber of student s left without a seat can be for- m ulated us ing a single term in the ob jective function: X r ∈ R X p ∈ P X c ∈ C Students c > Capacity r T p,r,c (Studen ts c − C a pacit y r ) . (29) The seco nd key per fo rmance indicator , the num b er of missing days o f instruction summed across all cour ses, can be formulated us ing tw o a uxiliary arrays of decis io n v aria ble s . The first bina ry array , U, is indexed with co urses and days. U c,d is set to one, if and only if there are some even ts of course c held o n day d . The other ar r a y of integers, V , is indexed with cour ses. V a lue V c is bounded b elow by zero and above by the num b er of days in a week and represents the num b er of days co ur se c is s hort o f its r ecommended days 18 Edmund K. Burke et al. of instruction. This enables addition of the constra in ts: X r ∈ R T p,r,c ≤ U c,d ∀ c ∈ C ∀ d ∈ D ∀ p ∈ Perio ds d (30) X r ∈ R X p ∈ Periods d T p,r,c ≥ U c,d ∀ c ∈ C ∀ d ∈ D (31) V c + X d ∈ D U c,d ≥ MinDays c ∀ c ∈ C . (32) The term P c ∈ C V c can then ea sily b e added to th e o b jective function. How ever, it is only the fo rm ulation o f the third key p e rformance in- dicator, the p enalty incurr ed by pa tterns o f dis tinct da ily timetables of individual or groups of students, tha t proves to have a decisiv e impact on the p erformance of formulations of Udine Course Timetabling (Burke et a l., 2008). The penalisa tion of patter ns in timetables w a s tra ditionally for m u- lated “by feature” ( Av ella & V a s il’ev, 200 5). In an auxilia ry binar y arr a y S indexed with curric ula, days and features , S u,d,f is set to one, if and only if feature f is present in the timetable fo r curr ic ulum u and day d . In the case of the p enalisation of e vents timetabled for a curriculum o utside of a single consecutive blo c k of t wo or more ev ents p er da y of four p erio ds, there are four c onstraints: ∀ u ∈ U ,d ∈ D , ∀h p 1 ,p 2 ,p 3 ,p 4 i∈ Periods d X c ∈ HasC u X r ∈ R (T p 1 ,r,c − T p 2 ,r,c ) ≤ S u,d, 1 (33) ∀ u ∈ U ,d ∈ D , ∀h p 1 ,p 2 ,p 3 ,p 4 i∈ Periods d X c ∈ HasC u X r ∈ R (T p 2 ,r,c − T p 1 ,r,c − T p 3 ,r,c ) ≤ S u,d, 2 (34) ∀ u ∈ U ,d ∈ D , ∀h p 1 ,p 2 ,p 3 ,p 4 i∈ Periods d X c ∈ HasC u X r ∈ R (T p 3 ,r,c − T p 2 ,r,c − T p 4 ,r,c ) ≤ S u,d, 3 (35) ∀ u ∈ U ,d ∈ D , ∀h p 1 ,p 2 ,p 3 ,p 4 i∈ Periods d X c ∈ HasC u X r ∈ R (T p 4 ,r,c − T p 3 ,r,c ) ≤ S u,d, 4 (36) How ever, considerable improv ement in the p erformance of pattern p enal- isation can b e gained by in tro ducing the concept of the enumeration of patterns. It is o b viously p ossible to pre-co mpute a s et B o f n + 2 tuples w, x 1 , . . . , x n , m , where n is the num ber o f p erio ds p er day , x i is one if there is instruction in perio d i of the daily pattern and min us one otherwis e , w Sup ernod al F orm ulation of Graph Colouring 19 is the p enalt y attached to the pa ttern, a nd m is the sum o f p ositive v alues x i in the patterns decremented by one. Burk e et a l. (2008) ha ve studied a nu mber o f po ssible applications of this co ncept, with one of the b est p er- forming b eing the addition of cons tr ain ts, s uc h as in the c a se of four p erio ds per da y: ∀h w,x 1 ,x 2 ,x 3 ,x 4 ,m i∈ B ∀ u ∈ U ∀ d ∈ D ∀h p 1 ,p 2 ,p 3 ,p 4 i∈ Periods d w ( x 1 X c ∈ HasC u X r ∈ R T p 1 ,r,c + x 2 X c ∈ HasC u X r ∈ R T p 2 ,r,c + x 3 X c ∈ HasC u X r ∈ R T p 3 ,r,c + x 4 X c ∈ HasC u X r ∈ R T p 4 ,r,c − m ) ≤ X s ∈ Checks S u,d,s . (37) The third term in the ob jective function is P u ∈ U P d ∈ D P s ∈ Checks S u,d,s . F or further details on formulations o f these so ft constra in ts and their impact on the overall p e r formance, see Bur k e et al. (200 8). 5 Com putational Experi e nce In order to ev aluate per formance o f the new formulation, we have co nducted a num b er of exp erimen ts. W e rep ort: 1. the dimensions of reversible clique pa rtitions o btained from g r aphs in the standard DIMACS b enchmark 2. per formance gains on graph colour ing instances or iginating from timetabling, bo th from real-life and randomly genera ted instances of the Udine Course Timetabling pr oblem 3. per formance gains on the the complete instances of Udine Course Timetabling problem, as compa red to the effects of sy mmetry breaking built into CPLEX. All rep orted results were meas ured on a des k top PC running Linux, equipp ed with tw o In tel Pen tium 4 pro cessor s clock ed at 3.2 0 GHz. ILOG CPLE X version 10.0 integer prog ramming solver was restricted to use only a sing le thread on a single pr oces s or. Default pa r ameter s e ttings were used, o utside of settings for symmetry break ing des c r ibed b elow and settings imp o sing the time limit of one hour on run time p er instance. DIMACS instance s descib ed by Johnson and T rick (199 6) were downloaded from the on-line rep ository 2 . F our r eal-life timetabling instances were taken fr om the b enc hmar k used by (Gasp ero & Sc haerf, 2003, 2 006) and eighteen mor e instances were ob- tained us ing a rando m instance g enerator 3 developed by the a uthors. Their dimensions are listed in T able 4 . In all instances, each cours e has one to six even ts p er week, with the average being three, each teacher teaches o ne or 2 Av ailable at http://mat .gsia.cmu.edu/CO L OR/ (No v 7, 2007) 3 Av ailable at http://cs. nott.ac.uk/ ~ jxm/timeta bling/generator/ ( N o v 7, 2007) 20 Edmund K. Burke et al. t wo cour ses totalling at one to six hours per w eek, and enrolments consist of less than ten ev ents p er week, on a verage. All instances were passed to CPLEX in LP format as genera ted from sourc es in Zimpl, the free alge br aic mo delling lang ua ge (K och, 2004 ), and are av a ilable o n-line in Zimpl for ma t. Instances in LP format, whose tota l size exceeds 1.3 GB, can b e also made av ailable up on req uest. V er ification of the res ults is th us p ossible with freely av ailable solvers, such as SCIP (Ach terb erg, 200 7). First, we hav e obta ined reversible c liq ue partions from DIMA CS gra phs. T o illustra te the effects of pr e - proc e ssing of the o riginal gr aph on the size o f the reversible clique par titio n, in T able 3, we list the sizes first without using any pr eproc e ssing (under Q ), as well as after some pre-pro cessing spe cific to g raph colouring , but not sp ecific to the transfor mation (under Q ′ ). This prepro cessing included: – Remo v al of vertices of deg ree less than a low er b ound on the chromatic nu mber – Remo v al of vertices connected t o all o ther vertices in the graph – Remo v al of each vertex u whose neighbougho o d is a subset of the neigh- bo urhoo d of a nother non-a djacen t vertex v . F or details of the pr e-pro cessing and the so urce co de used, plea se see the authors’ website 4 . Second, we e v aluated per formance of the s tandard for m ulation o f gr aph colouring intro duced in Sectio n 2.1 and p erformance of the fo rm ulation pro - po sed in Section 3 o n the gr aph colo ur ing comp onent of instances o f Udine Course Timetabling . (The complete constraint se t was used, but no ob jec- tive function.) Notice (in Sectio n 4.1 ) that b oth formulations use the sa me amount of infor mation o n cliques found in the co nflict gr aph, only e x pressed in terms of different decision v ariables . F rom the results rep orted in T a ble 5, it seems that with the ex ception of a single ra ndom insta nce (ra nd16) a nd one heavily constrained r eal-life instance (udine4), the pro posed formulation per forms co ns iderably b etter. Next, we compar ed pe rformance of the formulations of Udine Co urse Timetabling, differing only in the formulation of the underlying graph colour- ing component. Notice that the CP LEX run time nec e s sary to reach opti- mality was tw o orders of magnitude highe r than in the pr evious exp eriment lo oking for f ea s ible colouring. W hether the perfor mance ga ins observed in the g raph colo ur ing compo nen t alone would still b e manifested, was thus not cle a r. As is s umma r ised in T a ble 6 , how ever, the new formulation again seems to p erform c onsiderably b e tter , reducing CPLE X run times approxi- mately by factor of four, where it is possible to reach o ptim um within one hour using b oth formulations. W e have als o s tudied effects of symmetry breaking implemented in CPLE X on p erformance of b oth formulations. In all pr evious ex periments, b oth for - m ulatio ns were r un using no built-in symmetry breaking in CPLE X. T able 7 compares these results (denoted -SB) with results obtained with symmetry 4 Av ailable at http://cs. nott.ac.uk/ ~ jxm/colour ing/supernodal/ (No v 7, 2008) Sup ernod al F orm ulation of Graph Colouring 21 breaking built-in in CPLEX 10.0 set to aggressive (denoted + SB). Ag ain, the new form ulation using no built-in symmetry breaking performs b etter than the standa rd for m ulation us ing aggr e s siv e built-in s y mmetry breaking. These results ar e ra ther encouraging, although the p erformance gains are limited only to g raphs, where it is p ossible to obtain a reversible cliq ue partition of V , whose cardinalit y is consider ably less than | V | . This is not the cas e in tr iangle-free gr aphs and many dense r andom gr aphs, often used in b enc hmar k ing general graph colo uring pr ocedur es. In many rea l-w or ld applications, the graphs seem to b e, how ever, highly structured, a nd the structure is worth exploiting. 6 Concl usions and F urther W o rk W e hav e presented a transforma tion of graph co louring to g raph multi- colouring, mak ing it p ossible to use the standard for mulation of gra ph mul- ticolouring as a for m ulation of gra ph colo uring. This can also be view ed as the sup erno da l integer progr a mming formulation o f gr aph colo ur ing, where sup e rnode of Geor ge and McInt yre (197 8) is the complete subset of vertices of a g raph where each tw o vertices hav e the same neighbo ur s outside of the subset. It remains to be s e en, if the transformatio n could b e used in conjuction with other formulations of multicolouring. This transformation ca n b e seen as an example of symmetry breaking. Although there has b een recently a consider able interest (Mar g ot, 2002, 2003, 2 007; Ostrowski, Linderoth, Rossi, & Smriglio, 2007; K a ibel, Pein- hardt, & Pfetsch, 20 0 7; Kaib el & Pfetsch, 2008 ) in the developmen t of meth- o ds fo r automated s ymmetry breaking, these metho ds hav e so far not b een comp etitiv e in solving g raph colouring proble ms (Kaibel & Marg ot, 200 7). Compared to the sta nda rd formulation with symmetry breaking embed- ded in ILO G CPLEX 10.0 , the industr ial s tandard in integer progr amming solvers, our reformulation without the em b edded symmetry breaking en- abled offers p erformance , which is improv ed b y a factor of three. It would app ear that application-sp ecific fo r m ulations breaking symmetr ie s will be necessary , at least until p erformance of automated symmetry breaking im- prov es. Additionally , we hav e briefly survey ed s ev en other integer pro gramming formulations of vertex colour ing, prop osed in the litera ture. This seems to b e the first time such a survey has b een attempted. Genera lly sp eaking, in non- trivial applica tio ns of graph co louring, the p erformance o f v arious integer progra mming form ulatio ns of the underlying graph co louring comp onen ts seems to b e highly dep enden t on their suitability for a pplication-sp ecific key per formance indica tors. Nevertheless, a prop er co mputatio nal compar ison of int ege r prog ramming formulations of gra ph colo ur ing would b e mo st inter- esting – and remains to b e conducted. Another interesting resear ch dir e ction might ex plore hybridisation, us ing one enco ding in a n integer pro gramming formulation, but multip le enco dings for cut genera tion. 22 Edmund K. Burke et al. Finally , the pr opos e d formulation seems very co n venien t in timetabling applications. Compared t o many fo rm ulations necessitating column gener- ation, it is easy to extend this formulation to accommo date co mplex k ey per formance indicators (“soft constra in ts”). W e hav e demonstra ted its p er- formance o n the example of Udine Course Timeta bling, a b enc hmark pr ob- lem in timetabling with soft constra in ts pr opos ed by Gasp ero and Schaerf (2003). Using I L O G CPLEX 10 .0 , we have b een able to arrive at the previ- ously unknown o ptim um for instance Udine1 within 14 3 seconds on a single pro cessor. Such results give a new hop e that r eal-life instances of cour se timetabling could b e solv ed within prov ably small bo unds of o ptimalit y . A cknow le dgements. The autho r s ar e g r ateful to Andrea Schaerf and Luca Di Gaspero , who k indly maintain the Udine Co urse Timetabling problem, and to Ja y Y ellen, whose comments hav e help ed to impro ve a n early dra ft of the pa per. The comments of tw o anonymous refer ees, which help ed to improv e the pr e sen tation, ar e also muc h apprecia ted. Ha na Rudov´ a is in part supp orted by M ˇ SMT Pr o ject MSM002 1622419 and GA ˇ CR Pr o ject No. 201/0 7/0205. Andrew J. Parkes was supp orted b y EP SR C gra n t GR/ T26115/0 1. Sup ernod al F orm ulation of Graph Colouring 23 References Aardal, K. I., Ho esel, S. P. M. v an, Koster, A. M. C. A., & Mannino, C. (2007). Mo dels and solution techniques for frequency assignmen t problems. A nn . O p er. R es. , 153 , 79–129. Ac hliopta s, D., & Nao r, A. (2005). The tw o p ossible v alues of the chromatic nu mber of a ra ndo m gr aph. Ann. of Ma th. , 1 63 (3), 133 3–1349. Ac hterber g, T. (2007). Constr aint inte ger pr o gr amming . Unpublished do c- toral diss ertation, Ber lin. Appa, G., Ma gos, D., & Mourtos, I. (2005). On the system of tw o all different predicates. Inform. Pr o c ess. L ett . , 94 (3), 99–105. Avella, P ., & V a s il’ev, I. (2005). 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Zuck erman, D. (20 07). Linear degr ee extr actors and the inappr o x- imability of max cliq ue and chromatic num ber . The ory Comput. , 3 (6), 103 –128. Av ailable fro m http:// www.theo ryofcomputing .org/a rticles/ main/v003/a006 28 Edmund K. Burke et al. T able 3: Dimensio ns of gra phs induced by reversible clique partitions ob- tained from DIMACS instances ( G ), with ( Q ′ ) and without ( Q ) pre- pro cessing. Empty spaces indicate gra phs tr ivial to co lo ur. Instance Original Graph G Rev. Cliq . Part. Q Rev. Cliq. Part. Q ′ V e rt. Edges V ert. Edges V ert. Edges 1-F ullIns 3 30 100 29 89 1-F ullIns 4 93 593 92 561 2 5 85 1-F ullIns 5 282 324 7 281 3152 61 358 1-Insertions 4 67 232 67 232 6 0 208 1-Insertions 5 202 1 227 2 02 1227 202 122 7 1-Insertions 6 607 6 337 6 07 6337 600 630 1 2-F ullIns 3 52 201 51 186 2-F ullIns 4 212 162 1 211 1566 16 6 5 2-F ullIns 5 852 1 2201 85 1 11986 93 582 2-Insertions 3 37 72 37 72 2-Insertions 4 149 541 149 541 14 9 541 2-Insertions 5 597 3 936 5 97 3936 597 3936 3-F ullIns 3 80 346 79 327 1 7 65 3-F ullIns 4 405 352 4 404 3440 22 114 3-F ullIns 5 2030 33751 2029 33342 94 76 8 3-Insertions 3 56 110 56 110 3-Insertions 4 281 1 046 2 81 1046 281 1046 3-Insertions 5 1406 96 95 1406 9695 1395 9642 4-F ullIns 3 114 5 41 113 518 4-F ullIns 4 690 665 0 689 6531 4-F ullIns 5 4146 77305 4145 76610 1 95 176 9 4-Insertions 3 79 156 79 156 4-Insertions 4 475 1 795 4 75 1795 475 1795 5-F ullIns 3 154 7 92 153 765 39 229 5-F ullIns 4 1085 11395 1084 11235 1 21 103 7 abb313GP IA 1557 5 3356 1557 53356 853 16093 anna 138 493 125 437 ash331 GPIA 662 418 1 662 4181 661 4180 ash608 GPIA 121 6 7 844 1216 784 4 121 5 7843 ash958 GPIA 191 6 12506 1916 12 506 1 915 12505 david 87 406 74 322 DSJC1000.1 1000 49629 100 0 49 629 1 000 49629 DSJC1000.5 1000 24982 6 1 000 249826 1000 24982 6 DSJC1000.9 1000 44944 9 1 000 449449 1000 44944 9 DSJC125.1 125 7 36 125 736 125 7 36 DSJC125.5 125 389 1 125 3891 125 3891 DSJC125.9 125 696 1 125 6961 125 6961 DSJC250.1 250 321 8 250 3218 250 3218 DSJC250.5 250 1 5668 25 0 15668 250 15668 DSJC250.9 250 2 7897 25 0 27897 250 27897 Sup ernod al F orm ulation of Graph Colouring 29 T able 3: Dimensio ns of gra phs induced by reversible clique partitions ob- tained fro m DIMA CS instances. (Contin ued.) Instance Original Graph G Rev. Cliq . Part. Q Rev. Cliq. Part. Q ′ V e rt. Edges V ert. Edges V ert. Edges DSJC500.1 500 1 2458 50 0 12458 500 12458 DSJC500.5 500 6 2624 50 0 62624 500 62624 DSJC500.9 500 112437 500 11 2437 50 0 11 2437 DSJR500.1 500 3555 480 3341 DSJR500.1c 500 121275 500 12 1275 28 1 38166 DSJR500.5 500 588 6 2 497 58 218 483 5661 8 ear 190 479 3 185 4758 172 4636 fpsol2.i.1 496 1 1654 42 7 5108 107 2454 fpsol2.i.2 451 869 1 395 5657 154 2705 fpsol2.i.3 425 868 8 369 5658 153 2665 games12 0 120 63 8 119 629 hec 8 1 1363 81 1363 75 1277 homer 561 162 8 503 1376 hu ck 74 301 54 179 inithx.i.1 8 64 1870 7 732 111 40 inithx.i.2 6 45 1397 9 539 93 17 5 0 544 inithx.i.3 6 21 1396 9 521 94 27 4 9 474 jean 8 0 254 6 7 1 77 latin square 10 900 3073 50 900 30 7350 90 0 30 7350 le450 1 5a 45 0 81 68 45 0 8 168 4 07 780 2 le450 1 5b 45 0 81 69 45 0 8 169 4 10 782 4 le450 1 5c 450 1 6680 45 0 16680 450 16680 le450 1 5d 45 0 16750 450 16 7 50 450 1675 0 le450 2 5a 45 0 82 60 45 0 8 260 2 64 584 0 le450 2 5b 45 0 82 63 45 0 8 263 2 94 624 0 le450 2 5c 450 1 7343 45 0 17343 435 17096 le450 2 5d 45 0 17425 450 17 4 25 433 1710 6 le450 5 a 450 571 4 450 5714 450 5714 le450 5 b 450 5 734 4 50 5734 450 5734 le450 5 c 450 980 3 450 9803 450 9803 le450 5 d 450 9 757 4 50 9757 450 9757 miles1000 128 3216 123 3 049 miles1500 128 5198 104 3 486 miles250 128 3 87 117 341 miles500 128 117 0 115 1065 miles750 128 211 3 122 2011 m ug1 00 1 100 166 8 4 118 m ug1 00 25 100 166 83 115 m ug8 8 1 88 146 75 107 m ug8 8 2 5 88 146 7 2 9 8 m ulso l.i.1 197 39 25 16 6 2 2 74 30 Edmund K. Burke et al. T able 3: Dimensio ns of gra phs induced by reversible clique partitions ob- tained fro m DIMA CS instances. (Contin ued.) Instance Original Graph G Rev. Cliq . Part. Q Rev. Cliq. Part. Q ′ V e rt. Edges V ert. Edges V ert. Edges m ulso l.i.2 188 38 85 15 8 2 4 58 35 337 m ulso l.i.3 184 39 16 15 5 2 5 04 35 336 m ulso l.i.4 185 39 46 15 5 2 5 04 36 360 m ulso l.i.5 186 39 73 15 7 2 5 49 36 356 m yciel2 m yciel3 11 20 11 20 1 1 20 m yciel4 23 71 23 71 2 3 71 m yciel5 47 236 47 236 4 7 236 m yciel6 95 755 95 755 9 5 755 m yciel7 191 236 0 191 2360 191 2360 qg.order 100 10000 990000 100 00 990000 1000 0 990 000 qg.order 30 90 0 26100 900 26 1 00 900 2610 0 qg.order 40 1 6 00 6240 0 1 600 62400 1600 62 400 qg.order 60 3 6 00 212400 3600 21240 0 360 0 2 12400 queen10 10 100 14 70 10 0 1 470 1 00 147 0 queen11 11 121 19 80 12 1 1 980 1 21 198 0 queen12 12 144 25 96 14 4 2 596 1 44 259 6 queen13 13 169 33 28 16 9 3 328 1 69 332 8 queen14 14 196 41 86 19 6 4 186 1 96 418 6 queen15 15 225 51 80 22 5 5 180 2 25 518 0 queen16 16 256 63 20 25 6 6 320 2 56 632 0 queen5 5 25 160 25 160 2 5 160 queen6 6 36 290 36 290 3 6 290 queen7 7 49 476 49 476 4 9 476 queen8 1 2 96 136 8 96 1368 96 1368 queen8 8 64 728 64 728 6 4 728 queen9 9 81 10 56 81 105 6 81 105 6 school1 385 1909 5 376 18 937 353 187 99 school1 ns h 3 52 1461 2 344 14 4 86 322 1434 3 wap01a 2368 11087 1 1 594 73666 1594 73 666 wap02a 2464 11174 2 1 594 72498 1594 72 498 wap03a 4730 28672 2 3 716 224640 3716 22464 0 wap04a 5231 29490 2 3 814 221704 3814 22170 4 wap05a 905 4308 1 749 35 116 746 351 02 wap06a 947 4357 1 741 34 012 735 337 60 wap07a 1809 10336 8 1 611 91746 1609 91 698 wap08a 1870 10417 6 1 628 91140 1627 91 122 will199GPIA 70 1 67 72 70 1 6 772 6 6 0 583 6 zeroin.i.1 211 4100 182 2131 zeroin.i.2 211 3541 188 2187 zeroin.i.3 206 3540 183 2186 Sup ernod al F orm ulation of Graph Colouring 31 T able 4: The dimensions of test instances: num b ers of even ts, o ccupancy measured as the num ber o f even ts divided by the num b er o f av ailable time- place slots, and dimensions of the constraint matrices pro duced by for m u- lations of Udine Cour se Timetabling (v ariable s × cons train ts, non-zero s in constaint matrix). Instance Ev. Occ. Sta ndard (Non-zero) New (Non- zero) rand01 100 70% 1 5415 × 3194 469.35k 5398 × 41 76 188.34k rand02 100 7 0 15415 × 3 197 508.63 k 5398 × 4 179 188.38k rand03 100 7 0 15415 × 3 197 522.44 k 5398 × 4 179 199.47k rand04 200 7 0 60835 × 6 447 2.03M 210 02 × 8 444 794.63 k rand05 200 7 0 60830 × 6 416 1.94M 206 96 × 8 381 754.97 k rand06 200 7 0 60830 × 6 417 2.16M 206 96 × 8 382 814.10 k rand07 300 7 0 136 270 × 9 799 4 .29M 48 174 × 1 2907 1.7 6M rand08 300 7 0 136 260 × 9 729 4 .19M 47 262 × 1 2773 1.6 9M rand09 300 7 0 136 255 × 9 698 4 .46M 46 806 × 1 2710 1.7 4M rand11 100 8 0 12935 × 3 296 356.88 k 5097 × 4 406 159.66k rand12 100 8 0 12925 × 3 233 380.59 k 4835 × 4 279 160.43k rand13 200 8 0 50835 × 6 402 1.71M 176 52 × 8 399 664.51 k rand14 200 8 0 50840 × 6 427 1.56M 179 08 × 8 456 623.57 k rand15 200 8 0 50830 × 6 371 1.49M 173 96 × 8 336 606.71 k rand16 300 8 0 113 755 × 9 627 3 .92M 39 231 × 1 2639 1.4 9M rand17 300 8 0 113 770 × 9 726 3 .64M 40 374 × 1 2834 1.4 8M rand18 300 8 0 113 760 × 9 650 3 .66M 39 612 × 1 2694 1.4 6M udine1 207 8 6 50350 × 4 297 963.38 k 11756 × 539 3 280.62k udine2 223 9 3 54440 × 5 626 1.30M 134 52 × 6 889 378.48 k udine3 252 9 7 66940 × 7 883 2.20M 160 36 × 9 252 579.15 k udine4 250 10 0 64200 × 1206 0 3.70 M 155 0 5 × 13 678 915.37 k 32 Edmund K. Burke et al. T able 5: The p erformanc e of the standard and the prop osed (New) form u- lation of v ertex c o louring, mea s ured in run times of CPLEX and n um b ers of iterations p erformed with no built-in symmetry breaking (-0). The last column lists ratios of CPLEX run times. Instance Std-0 (Its.) New-0 ( Its.) Std-0 New-0 rand01 2.85s 1635 0.90s 9 31 3.16 rand02 2.99s 1666 0.94s 1106 3 .18 rand03 9.92s 5792 1.05s 1045 9 .45 rand04 99.48s 26317 5.18s 2802 19.2 0 rand05 73.72s 19802 33.49s 17467 2.20 rand06 83.78s 22537 40.35s 19836 2.08 rand07 216.08 s 358 21 86.4 4s 255 4 1 2.50 rand08 59.70s 10760 43.45s 13342 1.37 rand09 127.19 s 221 55 98.3 2s 257 8 2 1.29 rand11 3.80s 1761 1.51s 1194 2 .52 rand12 4.55s 2005 2.31s 1377 1 .97 rand13 95.67s 22851 47.94s 18957 2.00 rand14 45.25s 10544 6.64s 2629 6.81 rand15 30.77s 6799 6.89s 2685 4.47 rand16 114.32 s 116 03 27 5.44s 51518 0.4 2 rand17 251.15 s 331 85 14 4.93s 36949 1.7 3 rand18 160.25 s 216 86 13 8.04s 34461 1.1 6 udine1 23.2 3s 8082 4 .4 5s 33 70 5 .22 udine2 14.5 1s 4749 10.04s 4826 1.45 udine3 83.4 1s 168 07 17.2 5s 116 98 4.84 udine4 144.49 s 306 55 14 5.99s 30655 0.9 9 Sup ernod al F orm ulation of Graph Colouring 33 T able 6: The p erformance o f t wo for m ulations o f Udine Co urse Timetabling, differing only in the for m ulation of the underlying graph colour ing co mpo- nent : run times of CPLEX or gaps remaining after 1 hour o f solving and nu mbers of iterations p erformed with no built-in symmetry breaking (-0). The last column lists ratios of CPLEX run times, where optimality was reached within 1 hour using bo th formulations. Instance Std-0 (Its.) New-0 ( Its.) Std-0 New-0 rand01 385.59s 180 8 54 8 4.42s 4373 7 4.57 rand02 290.09s 71537 72.4 2s 3 4296 4.01 rand03 443.95s 148 9 61 5 9.99s 2331 0 7.40 rand04 gap 0.24% 4199 10 1242.50 s 2101 04 rand05 gap 4.15% 3608 68 1194.71 s 2501 48 rand06 gap 8.33% 2999 98 1257.72 s 2470 75 rand07 gap 89.71% 2340 87 gap 9 0.11% 242978 rand08 gap 99.85% 2372 43 gap 9 9.90% 312158 rand09 gap 93.97% 1996 19 gap 9 5.44% 263820 rand10 285.91s 66842 70.1 7s 2 7416 4.07 rand11 211.71s 68244 61.3 2s 3 1738 3.45 rand12 337.31s 129 7 88 8 4.16s 4840 1 4.01 rand13 gap 0.24% 4311 48 884.60s 175513 rand14 gap 6.47% 3220 73 1356.97 s 3201 29 rand15 gap 1.74% 3035 18 1166.50 s 2807 22 rand16 gap 66.44% 1757 66 gap 6 7.19% 417706 rand17 gap 94.15% 2395 76 gap 9 4.06% 293519 rand18 gap 90.57% 2518 22 gap 4 9.34% 345817 udine1 1175.40s 1665 39 2 37.12s 10422 1 4.96 udine2 gap 100 .00% 63 9068 gap 100 .00% 331883 8 udine3 gap 99 .3 1% 36 7505 gap 59.5 9% 20 00062 udine4 gap 99 .6 9% 22 0364 gap infinite 9628 56 34 Edmund K. Burke et al. T able 7: The p erforma nce of t wo formulations of Udine Cours e Timetabling, differing only in the formulation o f the underlying g raph colour ing com- po nen t, and effects of disabling (+0 ) the built-in symmetry br eaking in CPLEX, or s etting it to very agg ressive (+3): run times o f CPL E X or gaps remaining after 1 hour of solving. Instance Std+0 New+0 Std+3 New+3 Std+3 New-0 rand01 385.59s 8 4.42s 165 .52s 76.4 5s 1.96 rand02 290.09s 7 2.42s 343 .33s 65.5 1s 4.74 rand03 443.95s 5 9.99s 298 .52s 72.0 6s 4.98 rand04 gap 0.2 4% 1242.50s gap 0 .24% 1356.63 s rand05 gap 4.1 5% 1194.71s gap 4 .15% 1107.12 s rand06 gap 8.3 3% 1257.72s gap 8 .33% 1162.52 s rand07 gap 89.71% gap 90.11% gap 89.71% gap 90.11% rand08 gap 99.85% gap 99.90% gap 99.85% gap 99.90% rand09 gap 93.97% gap 95.44% gap 93.97% gap 95.44% rand10 285.91s 7 0.17s 321 .51s 81.1 2s 4.58 rand11 211.71s 6 1.32s 207 .41s 56.7 9s 3.38 rand12 337.31s 8 4.16s 253 .75s 84.6 4s 3.02 rand13 gap 0.2 4% 884.60 s ga p 1.85% 79 5 .50s rand14 gap 6.4 7% 1356.97s gap 6 .47% 1197.39 s rand15 gap 1.7 4% 1166.50s gap 30.43 % 1051.74s rand16 gap 66.44% gap 67.19% gap 66.44% gap 67.19% rand17 gap 94.15% gap 94.06% gap 94.15% gap 94.06% rand18 gap 90.57% gap 49.34% gap 90.57% gap 92.25% udine1 1 175.40s 237 .12s 1247.33 s 142.84s 5.26 udine2 gap 100.00 % gap 100.0 0% g ap 100.00 % ga p 100.0 0% udine3 gap 99 .3 1% g ap 59.5 9% g a p 99.33 % gap 7 0.04% udine4 gap 99 .6 9% gap infinite gap infinite gap infinite