Reformulating global constraints: the Slide and Regular constraints

Transcription

1 Reformulting glol constrints: the Slide nd Regulr constrints Christin Bessiere 1, Emmnuel Herrd 2, Brhim Hnich 3, Zeynep Kiziltn 4, Clude-Guy Quimper 5, nd Toy Wlsh 6 1 LIRMM, University of Montpellier, Frnce. essiere@lirmm.fr 2 Emmnuel Herrd, 4C, University College Cork, Irelnd. eherrd@4c.ucc.ie 3 Brhim Hnich, Fculty of Computer Science, Izmir University of Economics, Turkey. rhim.hnich@ieu.edu.tr 4 Zeynep Kiziltn, Deprtment of Computer Science, University of Bologn, Itly. zeynep@cs.unio.it 5 Clude-Guy Quimper, Omeg Optimistion, Cnd. quimper@lumni.uwterloo.c 6 Toy Wlsh NICTA nd University of New South Wles, Sydney, Austrli. tw@cse.unsw.edu.u Astrct. Glol constrints re useful for modelling nd resoning out rel-world comintoril prolems. Unfortuntely, developing propgtion lgorithms to reson out glol constrints efficiently nd effectively is usully difficult nd complex process. In this pper, we show tht reformultion my e helpful in uilding such propgtors. We consider oth hrd nd soft forms of two powerful glol constrints, Slide nd Regulr. These glol constrints re useful to represent wide rnge of prolems like rostering nd scheduling where we hve sequence of decision vriles nd some constrint tht holds long the sequence. We show tht the different forms of Slide nd Regulr cn ll e reformulted s ech other. We lso show tht reformultion is n effective method to incorporte such glol constrints within n existing constrint toolkit. Finlly, this study provides insight into the close reltionship etween these two importnt glol constrints. 1 Introduction Glol constrints re one of the most importnt fetures of constrint progrmming. Glol constrints cpture common ptterns occurring in models of complex, rel-life comintoril prolems. For instnce, common pttern in rostering prolems is tht sequences of night shifts must e followed y severl dys off, nd tht no one is llowed to work more thn certin numer of consecutive night shifts. Such ptterns cn e modelled using Regulr constrint [6]. More recently, we hve proposed the glol Slide to model wide rnge of ptterns ppering in sequencing nd other prolems [3]. In this pper, we explore the reltionship etween these two glol constrints in depth. We show tht the Slide constrint cn e efficiently reformulted s the Regulr constrint nd vice vers.

2 In mny rel-world prolems, it is not possile to find fesile solution tht stisfies ll the constrints nd preferences of the user. Consider the prolem of llocting reviewers for ppers sumitted to conference. Typiclly, pper must e reviewed y certin numer of reviewers nd ech reviewer must hve certin numer of ppers. Reviewers lso indicte preferences over the ppers tht they would e hppy to review. Finding n lloction tht stisfies the ssignment constrints s well s ll the reviewer preferences my not e possile. We my however give n dditionl pper to some reviewers nd my ssign pper to reviewer even if she is not enthusistic out it, s long s she did not indicte conflict of interest. Soft versions of glol constrints re useful to model nd solve such over-constrined prolems. A recent direction is to convert over-constrined prolems into constrint optimiztion prolems y treting constrint violtions s costs. To reson out such prolems, we cn design specific cost-sed propgtors. Severl such soft glol constrints hve een proposed to model nd solve over-constrined prolems effectively nd efficiently (e.g., [7, 9, 10, 12]). Whilst efficient propgtors hve een developed for oth the Slide nd Regulr constrint nd re ville in numer of solvers, there hs een less work out soft versions of these constrints. For exmple, no GAC propgtors hve yet een developed for soft versions of the Slide constrint. One of our contriutions in this pper is to propose the first such propgtors. We show tht reformultion is n ttrctive mechnism lso to implement soft versions of the Slide nd Regulr glol constrints. This rest of this pper is structured s follows. After giving the necessry forml ckground in Section 2, we explin in detil in Section 3 the Slide nd Regulr constrints. Then we show in Section 4 how Slide cn e efficiently reformulted s the Regulr constrint nd vice vers. In Section 5, we focus on the soft versions of these glol constrints nd in Section 6 demonstrte tht the different types of SoftSlide constrints cn e reformulted s hrd forms of the Slide or soft form of the Regulr constrints. We provide experimentl proof in Section 7 tht reformulting glol constrints could e useful. We report relted work in Section 8 nd conclude in Section 9. 2 Forml Bckground A constrint stisfction prolem consists of set of vriles, ech with finite domin of vlues, nd set of constrints specifying llowed comintions of vlues for some suset of vriles. We consider finite domin integer vriles, nd use cpitl letters for vriles (e.g. X) nd lower cse for vlues (e.g. d). We write D(X) for the domin of vrile X. A constrint C defined on vriles [X i,..., X j ] is indicted s C(X i,..., X j ). Constrint solvers typiclly explore prtil ssignments enforcing locl consistency property using either specilised or generl purpose lgorithms. A constrint C is generlised rc consistent (GAC ) iff when vrile is ssigned ny of the vlues in its domin, there exist comptile vlues in the domins of ll the

3 other vriles of C. For inry constrints, generlised rc consistency is often clled simply rc consistency (AC). A constrint C is ound consistent (BC ) iff when vrile is ssigned the mximum (or minimum) vlue in its domin, there exist comptile vlues etween the mximum nd minimum domin vlue for ll the other vriles of C. A deterministic finite utomton (DFA) is descried y 5-tuple A = Q, Q F, q 0, δ, Σ where Σ is n lphet, Q is set of sttes, q 0 Q is the initil stte, Q F Q is set of finl sttes, nd δ Q Σ Q is trnsition tle. A sequence s 1,..., s n is ccepted y the utomton A if there exists sequence of sttes t 0,..., t n such tht t 0 = q 0 is the initil stte, t n Q F is finl stte, nd t i 1, s i, t i δ is trnsition in the trnsition tle. A lnguge L Σ is (possily infinite) set of sequences tken from n lphet Σ. A regulr lnguge is the set of sequences ccepted y DFA. The Hmming distnce etween two strings s 1 nd s 2 of the sme length is the numer of positions in which they differ. The Hmming distnce is denoted y H(s 1, s 2 ). For exmple, the Hmming distnce etween the strings c nd dc is 1 s they differ only on the second position. The Hmming distnce etween string s nd lnguge L is min{h(s, t) t L}. If L does not contin ny string of the sme length s s, the distnce etween s nd L is undefined. The edit distnce etween two strings s 1 nd s 2 is the minimum numer of chrcter insertions, deletions, nd replcements to string s 1 in order to otin s 2. The edit distnce is denoted y E(s 1, s 2 ). For exmple, the edit distnce etween the strings c nd dc is 2 s we cn replce the chrcter of the string c y nd insert d efore c to otin dc. The edit distnce etween string s nd lnguge L is min{e(s, t) t L}. If L is empty, the distnce etween s nd L is undefined. 3 Slide nd Regulr constrints The glol Regulr constrint ws introduced y Pesnt to model prolems in scheduling nd rostering [6]. The constrint is specified in terms of finite utomton which ccepts the string of vlues spelled out y the sequence of vriles. More precisely, Regulr(A, [X 1,..., X n ]) holds iff X 1 to X n form string ccepted y the DFA A. Such glol constrint cn e used to ensure certin ptterns do (or do not) occur over time. For exmple, in shift rostering, we might hve tht we cnnot work more thn three night shifts in row nd once sequence of night shifts ends, we must hve t lest two dys off. This cn esily e specified using finite utomton. We hve finite utomton A with the sttes n 1, n 2, n 3, o 1 nd ny. The trnsition tle is: ny, dy, ny, ny, night, n 1, ny, off, ny, n i, night, n i+1, n i, off, o 1, nd o 1, off, ny. For instnce, if we re in stte ny, we go to stte n 1 with input night. Pesnt gives propgtor for the Regulr constrint sed on dynmic progrmming which chieves GAC in O(nd Q ) time where Q is the numer of sttes of the utomton [6]. More recently, we introduced the glol Slide constrint [3]. We egin with its simplest form. If C is constrint of rity k then Slide(C, [X 1,..., X n ]) holds

4 iff C(X i,..., X i+k 1 ) itself holds for 1 i n k + 1. Tht is, we slide the constrint C down the sequence of vriles, X 1 to X n. For exmple, consider the cr sequencing prolem (pro001 in CSPLi) where we need to decide the order in which to uild crs on n ssemly line. We might wnt to ensure tht no more thn one out of every two crs hs the sun roof option s it tkes extr time to fit sun roof. This cn esily e specified with Slide constrint. We slide inry constrint down sequence of decision vriles representing the order in which crs will e produced. This inry constrint ensures tht one of or oth of the vriles within its scope does not represent cr with the sun roof option. A vrition of the dul encoding cn e used to mintin GAC on such Slide constrint in O(nkd k ) time [3]. A more complex form of the Slide constrint permits us to slide down two or more sequences of vriles t the sme time. For instnce, if C is constrint of rity 2k then Slide(C, [X 1,..., X n ], [Y 1,..., Y n ]) holds iff C(X i,..., X i+k 1, Y i,..., Y i+k 1, ) itself holds for 1 i n k + 1. Tht is, we slide the constrint C down the two sequences of vriles. Consider, for exmple, the glol contiguity constrint [5]. This ensures tht within sequence of 0/1 vriles, X 1 to X n, the 1 s occur in continuous lock. We cn model this y Slide constrint down two sequences of 0/1 vriles, X 1 to X n nd Y 1 to Y n. The second sequence of vriles, Y 1 to Y n record if we hve met the lock of 1 s yet or not. The constrint eing slid, C(X i, X i+1, Y i, Y i+1 ) holds iff (Y i = 0, X i = 0, X i+1 = Y i+1 ) or (Y i = Y i+1 = 1 nd X i X i+1 ). Such more complex forms of Slide cn e reformulted s the simple form of Slide down single sequence. We merely need to interleve the different sequences nd then slide suitly modified constrint over these interleved sequences. (See [3] for detils.) 4 Reformulting Slide nd Regulr We first show tht Slide nd Regulr cn e reformulted s ech other. As well s providing insight into the reltionship etween the two glol constrints, these reformultions will e useful in providing propgtors for soft versions of these glol constrints. 4.1 Regulr s Slide In [8], we give simple reformultion of the Regulr constrint in terms of Slide constrint. In ddition to the sequence of vriles long which the Regulr constrint is defined, we introduce sequence of vriles for the stte of the utomton. We then slide the trnsition reltion down the two sequences of vriles. For instnce, consider gin the shift rostering prolem from the lst section. We cn reformulte Regulr(A, [X 1,..., X n ]) s Slide(C, [X 1,..., X n+1 ], [Q 1,..., Q n ]) where X n+1 is dummy vrile, Q i re vriles representing the stte of the utomton, nd C(X i, X i+1, Q i, Q i+1 ) holds iff we move from stte Q i to Q i+1 on seeing X i. Oserve tht X i+1 is not used in the definition of C. We let it in its scope just to e consistent with

5 the simplified presenttion of Slide on multiple sequences tht we presented in Section 3. Enforcing GAC on this reformultion chieves GAC on the originl Regulr constrint. Hence, reformultion does not hinder propgtion. This reformultion is lso optiml in the sense tht, we cn enforce GAC on either the Regulr constrint or its reformultion into Slide in O(nd Q ) time. This complexity is lower thn the complexity of Slide in generl ecuse of the chrcteristics of the constrint eing slid (see [3] for detils). As we show in the experimentl section, this reformultion is lso prcticl mens to propgte the Regulr constrint. By reformulting Regulr s Slide constrint, we get n efficient nd incrementl propgtor tht cn outperform Pesnt s propgtor for Regulr sed on dynmic progrmming. 4.2 Slide s Regulr The reverse is lso possile. Tht is, we cn reformulte ny instnce of the Slide constrint using Regulr constrint. Agin this is optiml in the sense tht we cn enforce GAC on either the Slide constrint or the reformultion into Regulr in O(nkd k ) time. We prove this clim y constructing DFA A recognizing the lnguge ccepted y Slide constrint. Let Σ = n i=1 D(X i) e the lphet nd k e the rity of the constrint C. The sttes of A re given y the set of sequences Q = k 1 i=0 Σi. The empty sequence ɛ Q is the initil stte. Any sequence of length k 1 is finl stte. Let T = {[s 1,..., s k ] C([s 1,..., s k ])} e the set of sequences ccepted y the constrint C. We construct the trnsition tle δ of A s follows. Let w e sequence of length strictly smller thn k 1 nd c Σ e chrcter from the lphet. Let wc e the conctention of the sequence w nd the chrcter c. We hve trnsition w, c, wc δ if there exists sequence in T strting with wc. Let, Σ e two chrcters nd w Σ k 2 e sequence of length k 2 such tht w is sequence in T. Then we hve the trnsition w,, w δ. Notice tht stte w cn only e visited fter prsing the su-sequence w. Exmple 1. Consider the lphet Σ = {, } nd the constrint C tht ccepts ny sequence of length three ut the sequences nd. We otin the DFA depicted in Figure 1. The DFA A constructed in this wy represents the Slide constrint. Theorem 1. If n is greter thn or equl to the rity of C, then the lnguge {X 1... X n Slide(C, [X 1,..., X n ])} formed y the sequences stisfying the Slide constrint is equl to the set of sequences of length n recognized y the DFA A. Proof. We first prove tht every sequence s of length n ccepted y A is lso ccepted y the Slide constrint. Let w e the first k 1 chrcters in s. By construction of A fter reding these k 1 chrcters, the current stte is w nd there exists sequence in T strting with w. Assume tht the k 1 chrcters

6 ɛ Fig. 1. DFA corresponding to Exmple 1. following position i in s re given y w where is chrcter nd w is sequence of length k 2. Also ssume tht there exists sequence in T strting with w nd tht fter reding w, A is in stte w. By construction of A when reding the chrcter t position i, the stte of A chnges from w to w. The trnsition gurntees tht the sequence w elongs to T nd tht the constrint C is stisfied t position i. By inductively repeting the rgument, we conclude tht the constrint C is stisfied t every position nd tht consequently, the Slide constrint is stisfied. We now prove tht if the sequence s stisfies the Slide constrint, then it is ccepted y A. Let w e the first k chrcters of s where nd re two chrcters nd w is sequence of k 2 chrcters. Since the Slide constrint is stisfied, the sequence w stisfies the constrint C nd elongs to the set T. Therefore, there is series of sttes q 0,..., q k 1 tht prses the sequence w where stte q i is the first i chrcters of w. Suppose tht the i th chrcter to e prsed is, tht the lst k 1 prsed chrcters re the chrcter followed y the sequence w, nd tht A is in stte w. Since the Slide constrint is stisfied, the sequence w stisfies C. Therefore, there exists trnsition w,, w δ tht prses the chrcter nd leds to the stte w. Notice tht w re the lst k 1 prsed chrcters. By inductively repeting the rgument, we conclude tht there exists prsing for ny sequence stisfying the slide constrint. Consequently, the sequence s is ccepted y A.

7 5 Softening Slide nd Regulr As we discussed in the introduction, rel world prolems re often over-constrined. One mechnism to del with such over-constrined prolems is to introduce cost function, nd find solution of miniml cost from fesile solution. We will formlize this process s follows. Let C(X 1,..., X n ) e hrd glol constrint like Slide or Regulr. Given distnce function d etween strings, the soft constrint C soft (X 1,..., X n, Z) holds iff Z = min{d( 1,..., n, 1,..., m ) i D(X i ) i 1..n & C( 1,..., m )}. Distnce might, for instnce, e Hmming (in which cse n = m) or edit distnce. As stted in Section 2, d is not defined if the set of strings stisfying C is empty. As n exmple, the constrint SoftSlide H (C, [X 1,..., X n ], D) holds iff D is the Hmming-distnce etween the sequence [X 1,..., X n ] nd the lnguge ccepted y the Slide constrint. Similrly, the constrint SoftSlide E (C, [X 1,..., X n ], D) holds iff D is the edit-distnce etween the sequence [X 1,..., X n ] nd the lnguge ccepted y the Slide constrint. Similrly, SoftRegulr H nd SoftRegulr E re soft forms of Regulr otined y pplying Hmming nd edit distnce sed violtions to Regulr respectively [10]. For exmple, SoftRegulr E (A, [X 1,..., X n ], D) is stisfied iff D is the edit-distnce etween the sequence [X 1,..., X n ] nd the lnguge ccepted y A. In [10], the propgtion lgorithm for Regulr sed on dynmic progrmming is modified in two different wys to mintin GAC on SoftRegulr H nd SoftRegulr E, respectively. 6 SoftSlide constrint We will show tht the different types of SoftSlide constrints cn e reformulted s hrd forms of the Slide or soft form of the Regulr constrints. Reformultion thus provides simple mechnism to propgte the soft forms of these glol constrints. 6.1 SoftSlide H s Slide Let us consider SoftSlide H (C, [X 1,..., X n ], D) where the rity of constrint C is k nd where n k. In order to reformulte this SoftSlide constrint s Slide constrint, we introduce two sequences of extr vriles. The first sequence contins n+k 1 vriles [S 1,..., S n+k 1 ] tht we uild in such wy tht S 1,..., S n cn e ny string ccepted y Slide(C, [S 1,..., S n ]). The domin of ech S i contins ll the vlues tht pper in t lest one tuple elonging to C. The vriles S n+1 to S n+k 1 must not e forced to stisfy the constrint eing slid. Hence, dummy vlue is dded to the domin of every S i, i > n. Furthermore, C is defined s relxtion of C tht contins tuple t iff t elongs to C or t contins t lest one dummy vlue. For instnce, if C is ternry constrint llowing the following set of tuples {,,,, c,,,, c },

8 then the domin of ech S i, i n, is {,, c}, the domin of ech S i, i > n, is {,, c, }, nd C = C { d 1, d 2, d 3 i 1..3, d i = }. The second sequence of vriles contins n + 1 vriles [D 1,..., D n+1 ] which provide the cumultive count of the numer of discrepncies with respect to the previous sequence. Then we introduce the following k + 3-ry constrint: C H (X i, S i,..., S i+k 1, D i, D i+1 ) C (S i,..., S i+k 1 ) D i+1 = D i + (S i X i ) This constrint ensures tht C is stisfied y the sequence [S i,..., S i+k 1 ] (tht is, [S i,..., S i+k 1 ] stisfies C if i + k 1 n), nd D i+1 = D i if S i = X i, otherwise D i+1 = D i + 1. Using the more complex form of Slide over multiple sequences, we cn thus reformulte the SoftSlide H constrint y sliding C H over the three sequences [S 1,..., S n+k 1 ], [X 1,..., X n ], nd [D 1,..., D n+1 ] nd y constrining D 1 to e 0 nd D n+1 to e equl to D. Tht is, we hve: SoftSlide H (C, [X 1,..., X n ], D) Slide(C H, [S 1,..., S n+k 1 ], [X 1,..., X n ], [0, D 2..., D n, D]) Enforcing GAC on Slide is in O(nkd k ), where n is the length of the sequence nd k the rity of the constrint eing slid. In the cse of the reformultion of SoftSlide H s Slide, the constrint to e slid hs rity k + 3. Thus, the time complexity of enforcing GAC on SoftSlide H using this reformultion is in O(nkd k+3 ) where d is the numer of vlues tht re used in tuples llowed y the constrint C. Note tht for this encoding to work correctly, the vriles [S 1,..., S n+k 1 ] should not e constrined y other constrints, so tht GAC on Slide(C, [S 1,..., S n+k 1 ]) gurntees solution. Since such vriles re introduced during the reformultion, they will e invisile to the users of the SoftSlide H constrint, hence such n ssumption is resonle. 6.2 SoftSlide E s SoftRegulr E By using the sme reformultion we proposed of Slide s Regulr, we cn reformulte SoftSlide E s SoftRegulr E. The set of strings ccepted y the hrd version of the SoftSlide E constrint must not e empty ecuse the propgtor of SoftRegulr E descried in [10] is defined for utomt ccepting non empty lnguge. This propgtor chieves GAC on the vriles X i for 1 i n nd BC on D in O(n δ + n Q log(n Q )) steps. The DFA A hs O( Σ i ) sttes lelled with sequence of length i for totl of Q = k 1 i=0 O( Σ i ) = O( Σ k ) sttes. The outgoing degree of every stte is ounded y Σ. We therefore hve δ = O( Σ k+1 ) trnsitions. Filtering the SoftSlide E constrint therefore requires O(n Σ k+1 + n Σ k log(n Σ k )) = O(n Σ k+1 + nk Σ k log(n Σ )) time.

9 7 Experimentl nlysis We now show tht reformultion is n effective mechnism to provide propgtors for Slide nd Regulr constrints. First, we compre resoning out Regulr constrint encoded s Slide, with resoning out it directly using Pesnt s GAC propgtor [6]. Then, we nlyse the reverse reformultion nd compre resoning Slide constrint reformulted s Regulr, with resoning out it directly using the GAC propgtor given in [3]. Pesnt presents two propgtors. We implemented the one tht keeps trck of ll supports for ech vlue in the domin of every vrile. Whilst the first set of experiments re done using ILOG Solver 6.1 on 900 MHz Pentium running Linux Dein, the second set is done using ILOG Solver 6.2 on 2.8GHz Intel Xeon computer running Linux FC Regulr s Slide As in [6], we generted rndom utomt with Q sttes nd n lphet of size Σ. We selected 30% of ll possile tuples (c, q i ) Σ Q nd rndomly chose stte q j Q to form the trnsition T (c, q i ) = q j. We otined the set of finl sttes F y rndomly selecting 50% of the sttes in Q. Following Pesnt, we used rndom vrile ordering nd rndom vlue selection. All experiments re verged over 30 runs. Tle 1 shows the results. We oserve tht the reformultion of Regulr constrints in terms of Slide is s efficient s nd most of the times slightly more efficient thn propgting directly the Regulr constrints. The propgtor for Slide uses sequence of uilt-in Tle constrints. We conjecture tht these re highly optimized nd contriute to the performnce offered y Slide. We lso rn experiments on model for the Mystery Shopper prolem due to Helmut Simonis tht ppers in CSPLi (pro004). This model contins lrge numer of Among constrints. We represented these Among constrints using Regulr constrints, nd gin either resoned with these Regulr constrints directly using Pesnt s propgtor or reformulted them using Slide constrints. Results re given in Tle 2. All instnces solved in the experiments use time limit of 5 minutes. Both methods chieve GAC on the Among constrint, so the serch trees re identicl nd it is only the efficiency of the propgtor which differ. Agin, reformultion of Regulr using Slide is slightly more efficient. 7.2 Slide s Regulr We consider vrint of the Nurse Scheduling Prolem [4] tht consists of generting schedule for ech nurse of shifts duties nd dys off within short-term plnning period. There re three types of shifts (dy, evening, nd night). We ensure tht (1) ech nurse should tke dy off or e ssigned to n ville shift; (2) ech shift hs minimum required numer of nurses; (3) ech nurse work lod should e etween specific lower nd upper ounds; (4) ech nurse

10 n Σ Q Regulr Regulr s Slide n Σ Q Regulr Regulr s Slide Tle 1. Time in seconds to find sequence stisfying rndomly generted utomton either using Pesnt s propgtor for the Regulr constrint or reformulting it s Slide constrint cn work t most 5 consecutive dys; (5) ech nurse must hve t lest 12 hours of rek etween two shifts; (6) the shift ssigned to nurse cnnot chnge more thn once every three dys. We develop two models to solve this prolem. In oth, we introduce one vrile for ech nurse nd ech dy, indicting to wht type of shift, if ny, this nurse is ffected on this dy. The constrints (1)-(3) re enforced using set of glol crdinlity constrints. The constrints (4), (5) nd (6) form sequences of respectively 6-ry, inry nd ternry constrints. Notice tht (4) is monotone, hence we simply post these constrints in oth models. The conjunction of constrints (5) nd (6) is slid using the tuple encoding of Slide in the first model, nd n encoding of Slide using Regulr in the second model. We test the models y using the instnces ville t in which nurses hve no mximum worklod, ut set of preferences is to e optimised. We ignore these preferences nd post constrint for ounding the mximum worklod to t most 5 dy shifts, 4 evening shifts nd 2 night shifts per nurses nd per week. Similrly, ech nurse must hve

11 Regulr Regulr s Slide Size #fils cpu time #solved #fils cpu time #solved / / / / / / / / / / / /56 Tle 2. Mystery Shopper prolem, Regulr v. Regulr s Slide. #fils nd cpu time re only verged on instnces solved y oth methods. Slide Slide s Regulr instnces solved 56/99 56/99 time cktrcks Tle 3. Nurse scheduling prolem (30 nurses, 28 dys), Slide v. Slide s Regulr. #fils nd cpu time re only verged on instnces solved y oth methods. t lest 2 rest dys per week. We solve smple of 99 instnces involving crew of 30 nurses to schedule over 28 dys. We use the sme sttic vrile ordering for oth models. The dys re scheduled in chronologicl order, nd within ech dy, we llocte shift to every nurse in lex order. Initil experiments show tht this simple heuristic is more efficient thn dynmic minimum domin heuristic. In Tle 3, we report the men fils nd cpu time required to solve the instnces. We oserve tht the first model is out 15% fster thn the second model. 8 Relted Work Reformulting new glol constrints in terms of those tht lredy ville within the constrint toolkit hs strted to gin ttention within the constrint progrmming community. For instnce, in [11], the AmongSeq constrint used in cr sequencing on production line is studied nd lterntive propgtion methods re discussed. One pproch reformultes AmongSeq s Regulr constrint. This reformultion is shown to e the most efficient in prctice compred to the other proposed propgtors. Given the lrge numer of glol constrints tht hve een identified, nother direction of study is generl-purpose glol constrints. Such constrints cn e used in conjunction with the primitive constrints to reformulte wide rnge of glol constrints without the need to extend the constrint toolkits. This is especilly useful if constrint toolkit does not provide propgtor for the glol constrint or if the constrint is difficult to propgte. Slide is such generl constrint ecuse it helps encode nd propgte mny sequencing

12 Regulr SoftSlide SoftRegulr Slide Fig. 2. The reltionship etween Slide, SoftSlide, Regulr, nd SoftRegulr constrints. constrints [3]. Other exmples of generl constrints re Rnge nd Roots [2]. They re shown to e very useful for reformulting diverse glol constrints ppering in counting nd occurrence prolems. One of the simplest wys to soften the Slide constrint is to relx the numer of times the slid constrint holds on the sequence. This gives the CrdPth constrint. CrdPth(C, [X 1,..., X n ], N) holds iff C holds N times on the sequence [X 1,..., X n ] [1]. Interestingly, the CrdPth constrint cn itself e reformulted s Slide constrint [3]. We cn therefore use the propgtor for Slide to propgte the CrdPth constrint. In fct, this reformultion is the first nd only method proposed so fr in the literture for enforcing GAC on CrdPth. 9 Conclusions To model rel-world constrint prolems nd to solve them efficiently, mny glol constrints hve een proposed in recent yers. In this pper, we hve focused on two importnt glol constrints, Slide nd Regulr which re useful for encoding nd propgting wide rnge of rostering nd sequencing prolems. Since prolems re often over-constrined, we hve lso studied soft forms of these glol constrints. We showed tht the different forms of Slide nd Regulr cn ll e reformulted s ech other. We lso showed tht reformultion is n effective method to incorporte such glol constrints within n existing constrint toolkit. This study hs provided insight into the close reltionship etween these two importnt glol constrints. The reltionships depicted in Figure 2 demonstrte the close links etween the hrd nd soft versions of the Slide nd Regulr constrints. An rrow from constrint C i to constrint C j indictes in the figure tht C i cn e reformulted s C j. A thick rrow is either due to findings in this pper or due to the fct tht soft form of constrint cn e used to propgte its hrd form y not llowing ny violtion. The dshed rrows cn e otined y trnsitivity

13 from the thick rrows. For instnce, given tht Regulr cn e reformulted s Slide which cn itself e reformulted s SoftSlide, we cn derive tht Regulr cn e reformulted s SoftSlide. References 1. N. Beldicenu nd M. Crlsson. Revisiting the crdinlity opertor nd introducing crdinlity-pth constrint fmily. In Proc. of ICLP 01, LNCS 2237,pp Springer, C. Bessière, E. Herrd, B. Hnich, Z. Kiziltn, nd T. Wlsh. The rnge nd roots constrints: Specifying counting nd occurrence prolems. In Proc. of IJCAI 05, pp Professionl Book Center, C. Bessière, E. Herrd, B. Hnich, Z. Kiziltn, nd T. Wlsh. The Slide Met- Constrint. Comic technicl report, 2006 (ville t hnich/comic/). 4. Burke, E. K., Cusmecker, P. D., Berghe, G. V., nd Lndeghem, H. V. The stte of the rt of nurse rostering. Journl of Scheduling, 7(6): , M. Mher. Anlysis of glol contiguity constrint. In Proc. of the CP 02 Workshop on Rule Bsed Constrint Resoning nd Progrmming, G. Pesnt. A regulr lnguge memership constrint for finite sequences of vriles. In Proc. of CP 04, LNCS 3258, pp Springer, T. Petit, J-C. Régin, nd C. Bessière. Specific filtering lgorithms for overconstrined prolems. In Proc. of CP 01, LNCS 2236, pp Springer, C.-G. Quimper nd T. Wlsh. Glol grmmr constrints. In Proc. of CP 06, LNCS 4204, pp Springer, W-J. vn Hoeve. A hyper-rc consistency lgorithm for the soft lldifferent constrint. In Proc. of CP 04, LNCS LNCS 3258, pp Springer, W-J. vn Hoeve, G. Pesnt, nd L-M. Rousseu. On glol wrming : Flow-sed soft glol constints. Journl of Heuristics, 12(4-5): , W-J. vn Hoeve, G. Pesnt, L-M. Rousseu, nd A. Shrwl. Revisiting the sequence constrint. In Proc. of CP 06, LNCS 42024, pp Springer, A. Znrini, M. Milno, nd G. Pesnt. Improved lgorithm for the soft glol crdinlity constrint. In Proc. of CP-AI-OR 06, LNCS 3990, pp Springer, 2006.