Towards Efficient Consistency Enforcement for Global Constraints in Weighted Constraint Satisfaction

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1 Towrs Effiient Consisteny Enforement for Glol Constrints in Weighte Constrint Stisftion J. H. M. Lee n K. L. Leung Deprtment of Computer Siene n Engineering The Chinese University of Hong Kong, Shtin, N.T., Hong Kong {jlee,klleung}@se.uhk.eu.hk Astrt Powerful onsisteny tehniques, suh s AC* n FDAC*, hve een evelope for Weighte Constrint Stisftion Prolems (WCSPs) to reue the spe in solution serh, ut re restrite to only unry n inry onstrints. On the other hn, vn Hoeve et l. evelope effiient grph-se lgorithms for hnling soft onstrints s lssil onstrint optimiztion prolems. We prove tht nively inorporting vn Hoeve s metho into the WCSP frmework n enfore strong form of -Inverse Consisteny, whih n prune infesile vlues n eue goo lower oun estimtes. We further show how Vn Hoeve s metho n e moifie so s to hnle ost projetion n extension to mintin the stronger AC* n FDAC* generlize for non-inry onstrints. Using the soft lldifferent onstrint s teste, preliminry results emonstrte tht our proposl gives improvements up to n orer of mgnitue oth in terms of time n pruning. Introution The tsk t hn is on how to relx or weken some of the hr onstrints in n over-onstrine prolem so s to otin useful prtil solutions. Weighte onstrint stisftion [Shiex et l., ] is frmework for hnling suh tsks. While the si tehnique for solving weighte onstrint stisftion prolems (WCSPs) relies on form of rnh-n-oun serh, vrious onsisteny notions n tehniques [Lrros n Shiex, 00; 00; Snhez et l., 00] for unry, inry, n ternry onstrints hve een evelope to help prune the serh spe. Higher rity onstrints hve to e either first onverte to their inry ounterprts or tivte only fter enough vriles re instntite uring serh. The lk of effiient hnling of noninry onstrints in WCSP systems gretly restrits the p- We thnk the nonymous referees for their onstrutive omments. The work esrie in this pper ws sustntilly supporte y grnts CUHK0 n CUHK0 from the Reserh Grnts Counil of Hong Kong SAR. pliility of WCSP tehniques to omplex rel-life prolems. Inorporting ritrry soft n-nry onstrints into WCSP n e iffiult sine the osts hve to e represente extensionlly n mintine in n n-imensionl tle, inurring time n spe overhes. Soft glol onstrints re non-inry onstrints with semntis. In prtiulr, the ost struture of flow-se soft glol onstrints [vn Hoeve et l., 00] n e formulte s flow network, llowing the omputtion of the minimum ost of the soft glol onstrints using minimum ost flow lgorithm. This is useful in estimting the lower oun of the urrent serh pth. We show tht nive inorportion of flow-se soft glol onstrints into WCSP woul result in strong form of the -inverse onsisteny [Zytniki et l., 00], whih is still reltively wek in terms of lower oun estimtion n pruning. The question eomes whether we n hieve stronger onsistenies, the generlize versions of AC* [Lrros n Shiex, 00] n FDAC* [Lrros n Shiex, 00], for non-inry onstrints effiiently. Consisteny lgorithms for AC* n FDAC* involve three min opertions: () omputing the minimum ost of the onstrint when vrile x is fixe with vlue v, () projeting the minimum ost of onstrint to the unry onstrint for x t vlue v, n () extening the unry ost to the non-unry onstrints. These opertions llow ost movement mong onstrints n shifting of ost to the C onstrint, resulting in higher lower oun n lso omin prunings. Prt () is reily hnle y the minimum ost flow (MCF) lgorithm. We show how the MCF lgorithm n the orresponing flow networks n e pte for prts () n () so s to perform projetion n extension in polynomil time n spe omplexity. Using the soft lldifferent onstrint s teste, we emonstrte the vntges of the stronger onsistenies over nive inorportion. WCSP The weighte CSP (WCSP) frmework extens lssil onstrint stisftion y ssoiting osts to the tuples of vrile ssignments. A WCSP [Shiex et l., ] is tuple (X, D, C, k). X is set of vriles {x,x,...,x n } orere y their inies. D is set of omins D(x i ) for x i X. Eh x i n only e ssigne one vlue in its orresponing omin. An ssignment on set of vriles n

2 e represente y tuple l. We enote l[x i ] s the vlue ssigne to x i,nl[s] s the tuple forme from the ssignment on suset of vriles S. C is set of onstrints, eh C S of whih represents funtion mpping tuples orresponing to ssignments on S to ost vlution struture V (k) =([0...k],, ). The struture V (k) ontins set of integers [0...k] with stnr integer orering. Aition is efine y = min(k, + ), n sutrtion is efine y = if k n k = k for ny. Without loss of generlity, C S n lwys e efine (initilly with ll tuples mpping to zero) for ll S X. Theost of tuple l orresponing to n ssignment on X is efine s: ost(l) =C C C S C S(l[S]), where C is null onstrint tht enotes the lower oun of osts of ll possile tuples. A tuple l orresponing to n ssignment on X is fesile if ost(l) <k,nissolution of WCSPifl hs the minimum ost mong ll fesile tuples. A soft glol onstrint C S on vriles S hs prtiulr semntis n n hve more thn one ost mesure. Where neessry, we sometimes give lso seprte ost funtion μ in se C S hs more thn one suh funtion. For simpliity, we ssume tht when we write C S (l) or μ(l), l is lwys fesile ssignment to S. WCSPs re solve with si rnh-n-oun serh ugmente with onsisteny tehniques whih prune infesile vlues from vrile omins n push lower oun estimtes into C. Common onsisteny notions n tehniques [Lrros n Shiex, 00; 00; Snhez et l., 00] inlue NC*, AC*, n FDAC*, ut re esigne for unry to ternry onstrints only. Avrilex i is NC* if () eh vlue v D(x i ) stisfies C xi (v) C <kn () there exists vlue v D(x i ) suh tht C xi (v )=0. A WCSP is NC* iff ll vriles re NC*. Algorithm enfores NC*. Funtion unryprojet() projets osts from unry onstrints to C y simple rithmeti opertions, n prunevl() removes infesile vlues from omins. Enforing IC n Strong IC In this setion, we explin how vn Hoeve s metho of using minimum ost flow n e pte for WCSPs to enfore strong form of -inverse onsisteny [Zytniki et l., 00]. Given onnete flow network G(V, E, w,, s, t), wherev re the verties, E re the eges, n eh ege e E hs weight w e n pity e.aflow f from soure s to sink t of vlue α in G is efine s mpping from E to R suh tht : (s,u) E f su = (u,t) E f ut = α; (u,v) E f uv = (v,u) E f vu v V \{s, t}; 0 f e e e E. If α is not efine, α is the mximum vlue of ll flows in G. The ost of flow f is efine s e E w ef e. A soft glol onstrint C S with ost funtion μ is flow-se if μ llows for representtion in flow network G so tht the flow with 0 Proeure enforenc*() foreh x i X o unryprojet(x i ); foreh x i X o prunevl(x i ); Proeure unryprojet(x i ) α := k; foreh v D(x i ) o if α>c xi (v) then α := C xi (v); C := C α; foreh v D(x i ) o C xi (v) :=C xi (v) α; return α>0; Funtion prunevl(x i ):Boolen foreh v D(x i ) o if C xi (v) C = k then D(x i ):=D(x i ) \{v}; Algorithm : Enfore NC* on WCSP minimum ost in G orrespons to the tuple mpping to the minimum ost in C S. Vn Hoeve et l. [00] emonstrte his frmework on the soft versions of the lldifferent, g, regulr,nsme onstrints. We use the soft lldifferent onstrint with the μ e ost mesure [Petit et l., 00] to illustrte the onepts. Given n ssignment tuple l on vriles S, μ e (l) = {(i, j) i <j l[x i ]=l[x j ] x i,x j S}, whih stns for the numer of pirs of vriles shring the sme vlue. We n onstrut flow network G(V, E, w,, s, t) s follows [vn Hoeve et l., 00]. The network onsists of S + D + noes, where D is the size of the union of ll vrile omins in S. Eh vrile n vlue hve n ssoite noe, with two more noes s n t. The network ontins three sets of eges: (s, x i ) E for eh x i X with zero weight n unit pity; (x i,v) E for eh v D(x i ) with zero weight n unit pity; (v, t) i E, for eh i =,..., v,where v is the numer of vriles ontining v. Eh ege (v, t) i hs unit pity n weight of i. For exmple, if X = {x,x,x,x } with D(x ) = {, }, D(x ) = {, },D(x ) = D(x ) = {, } n lldifferent(x), the network is onstrute s shown in Figure. Only non-zero weights re shown in the network. All eges ssume unit pity. The minimum ost of the fesile flow in G with vlue X is min{μ e (l)}. To ompute min{μ e (l) l[x] =v}, the minimum ost flow simply enfores f xv =. For exmple, Figure shows flow (highlighte y thikene eges) of minimum ost when f x =.Regin[00] n Vn Hoeve et l. [00] prove tht suh n enforement n e erive from n existing flow 0

3 s x x x x Figure : A flow network for lldifferent. The thik eges give the minimum ost flow when x =. y onstruting resiul network from G n the existing flow, n fining the minimum ost yle ontining (x, v) in the resiul network. This n e foun y using the single soure shortest pth lgorithm. We now efine IC [Zytniki et l., 00] n strong IC for WCSPs, the enforement of whih n enefit from vn Hoeve s metho. A onstrint C S is IC if there exists tuple l orresponing to fesile ssignment with C S (l) = 0. AWCSPis IC iff ll onstrints re IC. For exmple, Figure () shows WCSP whih is not IC. No mtter whih vlues re ssigne to the vriles, C x,x C =0 x x () not IC () IC Figure : Two equivlent WCSPs with k = C = x x returns ost of t lest. To enfore IC, ost of is projete iretly from C x,x to C y reuing the ost of eh tuple y n inrese C y. The resultnt WCSP isshowninfigure(). Proeure enfore IC() in Algorithm enfores IC for WCSP y enforing IC on eh onstrints. Reuing the ost of eh tuple n e expensive. An implementtion trik is to use zero-initilize vrile z S to store the ost reue so fr ue to projetion from C S to C. If tuple l queries its ost from C S, the result is C S (l) z S. In generl, the lgorithm is exponentil even with the implementtion trik sine exponentil numer of tuples hve to e exmine t line. However, minimum ost flow omputtion llows for polynomil time lgorithm for flow-se soft glol onstrints, suh s lldifferent with μ e. Enforing IC only inreses C. We oserve, for exmple in Figure (), tht the vlue D(x ) nnot e prt of ny solution. The tuple ssoite with x = hs ost t lest : from C, from C x,n from C x,x. Extr onitions n e e to strengthen IC to llow lso t Funtion enfore IC():Boolen foreh C S C o flg := flg enfore IC(C S ); Funtion enfore IC(C S ):Boolen α := min{c S (l)}; foreh tuple l o C S (l) :=C S (l) α; C := C α; return α>0 Algorithm : Enforing IC on WCSP omin reution. A non-unry onstrint C S is strong IC if: C S is IC, n; for ll vlues v D(x) with x S, C C x (v) min{c S (t) t[x] =v} <k. AWCSPisstrong IC iff ll onstrints re strong IC. For exmple, the WCSP in Figure () is not strong IC, ut removing the vlue from D(x ) mkes it so. Proeure enforestrong IC() in Algorithm enfores strong IC, se on the W-AC*() Algorithm [Lrros n Shiex, 00]. The lgorithm mintins propgtion queue Q (implemente s set) of vriles. Constrints involving vriles in Q re potentilly not strong IC. Funtion pop() removes n ritrry ville vrile from Q in onstnt time. 0 Proeure enforestrong IC() Q := X; while Q o x u := pop(q); foreh C S s.t. {x u } S o foreh x i S \{x u } o flg := removeinfesile(c S, x i ); if flg then Q := Q {x i }; if enfore IC() then Q := X; Funtion removeinfesile(c S, x i ):Boolen foreh v D(x i ) o α := min{c S (l) l[x i ]=v}; if C C xi (v) α = k then D(x i ):=D(x i ) \{v}; Algorithm : Enforing strong IC of WCSP

4 ProeureenforeStrong IC() in Algorithm must terminte, the proof of whih is similr to those of Lrros n Shiex s Theorems n [00]. Suppose removeinfesile() n enfore IC() hve time omplexity of O(f strong ) n O(f ) respetively, the omplexity n e stte s follows. Theorem Proeure enforestrong IC() hs time omplexity of O(s mx (s mx ef strong + f )), wheree is the numer of non-unry onstrints, s mx is the mximum rity of the onstrints, n is the numer of vriles, n is the mximum omin size. Thus, enforestrong IC() must terminte. Proof: In eh itertion of the while loop, line will e exeute O(s mx e) times. Eh vrile is pushe into Q t most O() times ue to line (eh time D(x u ) is moifie); thus the while loop will e exeute O(s mx ) times. Therefore, the omplexity of proeure enforestrong IC() is O(s mx (s mx ef strong + f )), n it must terminte. Agin, enforestrong IC() requires exponentil omplexity sine line is exponentil in generl. However, line n e ompute in polynomil time for flow-se soft glol onstrints. The following result is onsequene of Regin s Lemm [00] n vn Hoeve et l. s Theorem [00]. Theorem If C S is flow-se soft glol onstrint, removeinfesile() hs time omplexity of O(K + SP), whereo(k) n O(SP) re the time omplexity to fin the minimum ost flow n single soure shortest pth respetively, n is the mximum omin size. Given network G(V, E, w,, s, t). A typil O(K) is O( V E ) if the suessive shortest pth lgorithm is use, n typil O(SP) is O( V E ) if lel orreting lgorithm, like the Bellmn-For lgorithm, is use [Ahuj et l., 00]. Due to spe limittion, we nnot give the etils of the resoningthtthe SoftsHr pproh [Petit et l., 00] is slightly weker thn enforing strong IC together with NC*, whih is still reltively wek in terms of the eue lower oun n pruning. Stronger onsistenies for soft glol onstrints re esirle. Projetion in GAC* We speilize the efinition of GAC in Cooper et l. [00] for WCSP. A vrile x i S is generlize r onsistent str (GAC*) with respet to non-unry onstrint C S if: x i is NC*, n; for eh vlue v i D(x i ), there exists vlues v j D(x j ) for ll j i n x j S so tht they form tuple l with C S (l) =0. {v j } is simple support of v i with respet to C S. AWCSPisGAC* iff ll vriles re GAC* with respet to ll onstrints. Notie tht GAC* ollpses to AC* for inry onstrints [Lrros n Shiex, 00] n AC for ternry onstrints [Snhez et l., 00]. Proeure enforegac*() in Algorithm enfores GAC* for WCSP n is se on the W-AC*() Algorithm [Lrros n Shiex, 00]. Algorithm must termi- 0 0 Proeure enforegac*() Q := X; while Q o x u := pop(q); foreh C S s.t. {x u } S o foreh x i S \{x u } o flg := flg finsupport(c S, x i ); if prunevl(x i ) then Q := Q {x i }; if flg then foreh x i X o if prunevl(x i ) then Q := Q {x i }; Funtion finsupport(c S, x i ):Boolen foreh v D(x i ) o α := min{c S (l) l[x i ]=v}; if C xi (v) =0 α>0then C xi (v) :=C xi (v) α; foreh tuple l with l[x i ]=v o C S (l) :=C S (l) α; unryprojet(x i ); Algorithm : Enforing GAC* for WCSP nte, the proof of whih is similr to tht of Theorem. By repling O(f strong ) n O(f ) y O(f GAC ) (the omplexity of finsupport()) no(n) (n times the omplexity of prunevl()) respetively, the omplexity of Algorithm n e stte s follows. Theorem Proeure enforegac*() hs time omplexity of O(s mx (es mx f GAC + n)), wheren,, e, n s mx re s efine in Theorem. Thus, enforegac*() must terminte. Agin, Algorithm requires exponentil time omplexity sine funtion finsupport() is exponentil. The time omplexity of finsupport() is etermine y two opertions: minimum ost omputtion (line ) n ost projetion (lines to ). Line omputes the minimum ost of C S when x i = v. Line projets the ost to the unry onstrint C xi, whih is simple rithmeti opertion. Lines n upte the ost of ll tuples orresponing to x i = v. In generl, this two su-proeures require exponentil time omplexity, whih n e reue for flow-se soft glol onstrints. Vn Hoeve s metho n e pplie similrly to line s in Setion. Lines to moify the ost funtion of the soft (glol) onstrint C S. Before we give our metho, we stte the onitions uner whih our metho is pplile. A soft glol onstrint C S with ost funtion μ is projetion-sfe if the soft glol onstrint C S with ost funtion μ is flow-se, n hs the orresponing flow network

5 G(V, E, w,, s, t), there is one-one orresponene etween every flow f of G n omplete vrile ssignment tuple l for C S, n there exists n injetion from n ssignment x i = v to ē E suh tht whenever l[x i ] = v for some tuple l, fē =in the flow f orresponing to l; whenever l[x i ] v, fē =0. Given projetion-sfe soft glol onstrint C S with ost funtion μ efine ove. Suppose ost of α is projete from C S to C xi ssoite with x i = v, resulting in new ost funtion μ. In other wors, μ (l) = μ(l) α if l[x i ] = v; otherwiseμ (l) = μ(l). We onstrut the orresponing flow network of C S with ost funtion μ s G (V,E,w,,s,t),wherew e = w e α if e is the ege orresponing to x i = v; otherwisew e = w e. We use gin the lldifferent onstrint with μ e s n exmple. Figure shows the orresponing flow network n the flow representing (x,x,x,x )=(,,, ) with ost. If ost of is projete from the onstrint to C x ssoite with x =, new network n e onstrute y eresing the weight w x of the ege (x,) from 0 to, s shown in Figure. The new ost of the flow in the network is now 0, whih orrespons to the ost of the tuple (,,, ) fter projetion. s x x x x Figure : The flow network orresponing to lldifferent fter projetion. The sounness n losure of our metho re gurntee y the following theorem. Theorem Suppose C S is soft glol onstrint with ost funtion μ is projetion sfe, ost of α ssoite with x i = v is projete from C S to C xi, resulting in new ost funtion μ. (Sounness) If f is minimum ost flow of G (V,E,w,,s,t),then e E w ef e =min{μ (l)}. (Closure) C S with ost funtion μ is projetion sfe. Proof: Projetion-sfety implies tht e E w e f e = e E w ef e αfē =min{μ(l)} αfē =min{μ (l)},where ē is the ege orresponing to x i = v. This onlues sounness. In ition, C S with μ is flow-se with G (V,E,w,,s,t) s the orresponing flow network. Sine the topology of G (V,E,w,,s,t) is the sme s tht of G(V, E, w,, s, t), C S with μ is projetion-sfe. t We stte without proof tht the mjority of the flow-se glol onstrints [vn Hoeve et l., 00] re projetion-sfe so tht GAC* n e enfore on them in polynomil time. Theorem The following flow-se soft glol onstrints re projetion-sfe. lldifferent with either μ vr or μ e ; g with either μ vr or μ vl ; sme with μ vr ; Unfortuntely, the regulr onstrint with either μ vr or μ eit [vn Hoeve et l., 00] n the soft SEQUENCE onstrint [Mher et l., 00] re not projetion-sfe sine they o not stisfy the thir requirement. Agin, the omplexity of enforing GAC* for vrile with respet to the projetion-sfe soft glol onstrints follows from vn Hoeve et l. s Theorem. Theorem If C S is projetion sfe, finsupport() hs time omplexity of O(K + SP), wherek n SP re s efine in Theorem. Bse on FDAC* [Lrros n Shiex, 00], even stronger onsisteny n e efine ut its enforement involves n extension opertor, whih is the reverse of projetion n the fous of the next setion. Extension in FDGAC* Suppose vriles re orere y their inies. A vrile x i S is iretionl generlize r onsistent str (DGAC*) with respet to non-unry onstrint C S if: x i is NC*, n; for eh vlue v i D(x i ), there exists vlues v j D(x j ) for ll j i n x j S so tht they form tuple l with C S (l) j>i x j S C x j (v j )=0. {v j } is full support of v i with respet to C S. AWCSPisfull iretionl generlize r onsistent str (FDGAC*) if ll vriles re DGAC* n GAC* with respet to ll non-unry onstrints. When the onstrints re inry, FDGAC* ollpses to FDAC* [Lrros n Shiex, 00]. When the onstrints re inry n ternry, however, FDGAC* iffers slightly from FDAC [Snhez et l., 00]. FDGAC* requires full supports with only zero unry osts, while FDAC [Snhez et l., 00] requires full supports with not only zero unry ut lso zero inry osts. BseontheFDAC*() Algorithm [Lrros n Shiex, 00], proeure enforefdgac*() in Algorithm enfores FDGAC*. Q n R store vriles whih re potentilly not GAC* n not DGAC* respetively. Funtion popmx() lwys removes the vrile with the lrgest inex from R in onstnt time. Proeure enforefdgac*() in Algorithm must terminte, the proof of whih is similr to those of Lrros et l. s Theorems n [00]. Suppose finfullsupport() n finsupport() re of orer O(f DGAC ) n O(f GAC ) respetively, the omplexity of proeure enforefdgac*() n e stte s follows.

6 Proeure enforefdgac*() R := Q := X; while R Q o while Q o x u := pop(q); foreh C S s.t. {x u} S o foreh x i S \{x u} o if finsupport(c S, x i) then R := R {x i}; 0 0 if flg then foreh x i X s.t. prunevl(x i) o Q := Q {x i}; while R o x u := popmx(r); if prunevl(x u) then Q := Q {x u}; foreh C S s.t. {x u} S o for i = u DownTo s.t. x i S o if finfullsupport(c S, x i) then R := R {x i}; foreh x i X s.t. prunevl(x i) o Q := Q {x i}; Funtion finfullsupport(c S, x i):boolen foreh j>in x j S o foreh v D(x j) o foreh tuple l with l[x j]=v o C S(l) :=C S(l) C xj (v j); 0 C xj (v j):=0; flg := finsupport(c S, x i); foreh j>in x j S o finsupport(c S, x j); unryprojet(x i); Algorithm : Enforing FDGAC* on WCSP Theorem enforefdgac*() hs time omplexity of O(s mxe(nf DGAC + f GAC )+n ),wheren,, e, n s mx re efine in Theorem. Thus, enforefdgac*() must terminte. Agin, the omplexity n e exponentil ue to finsupport() n finfullsupport(). Inthefollowing, we fous the isussion on finfullsupport(). The first prt (lines to ) performs extension, reversl of projetion, to push ll the unry osts k to C S.Bythe time we exeute line, ll unry osts re 0, n enforing GAC* for x i hieves the seon requirement of DGAC*. Line re-insttes GAC* for ll vriles x j,wherej>i. Note tht the suess in line gurntees tht C xj (v j )=0 if v j ppers in tuple l whih mkes C S (l) =0. The key ie to performing extension properly is similr to tht of projetion: the metho is pplile to projetionsfe soft glol onstrint C S with ost funtion μ. Suppose now we wnt to exten ost of α ssoite with x i = v from C xi to C S resulting in new ost funtion μ.inother wors, μ (l) =μ(l) α if l[x i ]=v; otherwiseμ (l) = μ(l). We onstrut the orresponing flow network of C S with ost funtion μ s G (V,E,w,,s,t), wherew e = w e α if e is the ege orresponing to x i = v; otherwise w e = w e. Similrly, extension is oth soun n lose. Theorem Suppose C S is projetion sfe soft glol onstrint with ost funtion μ, n ost of α ssoite with x i = v is extene from C xi to C S, resulting in new ost funtion μ. (Sounness) If f is minimum ost flow of G (V,E,w,,s,t),then e E w e f e =min{μ (l)}. (Closure) C S with ost funtion μ is projetion sfe. The omplexity result gin follows from vn Hoeve et l. s Theorem [00]. Theorem If C S is projetion-sfe soft glol onstrint, finfullsupport() hs time omplexity of O(K + s mx SP),whereK n SP re s efine in Theorem. Lst ut not lest, we stte the reltive strength of the onsistenies onerne. Given two onsistenies β n γ, β is stronger thn γ (β γ)ifwcspp is γ whenever P is β. Theorem 0 FDGAC* GAC* strong IC GAC in Soft s Hr Approhes. Experimentl Results To emonstrte the effiieny of our proposls, we hve implemente strong IC, GAC*, n FDGAC* for the soft lldifferent onstrint with the μ e n μ vr ost funtions in ToulBr. Our enhmrk instnes re se on softene version of the ll-intervl series prolem (CSPLi Pro00). This prolem ontins minly lldifferent onstrints, unonerning us from other possile externl ftors n fousing on evluting the effiieny of our propose lgorithms. Suh enhmrk lso llows us to stuy the sling ehvior of our lgorithms. The originl prolem of orer n is to fin series {x,...,x n } suh tht it is permuttion of {0,...,n } n the jent ifferenes i = x i x i+, i = {,...,n } re istint. To moel its softene version s WCSP, we use {x i } n { i } s vriles with omins {0,...,n }. Two lldifferent onstrints re ple on {x i } n { i } respetively. Ternry tle onstrints re use to enfore i = x i x i+. Besies, rnom unry onstrints re ple on {x i }, ssigning rnom osts to eh ssignment rnging from 0 to. During the experiment, vriles {x i } re first ssigne in lexiogrphi orer, followe y { i } in the sme orer. Vlue ssignments strt with the vlue with minimum unry ost first. The test is onute in Dell Optiplex 0 with n Intel P.GHz CPU n GB RAM. The verge runtime n numer of ktrks of five instnes re mesure for eh vlue of n with no initil upper oun. Entries re mrke with * if the verge runtime exees the limit of hour.

7 n Strong IC GAC* FDGAC* Time(s) Bktrks Time(s) Bktrks Time(s) Bktrks * * * * * *.. * * * *.0. * * * *..0 * * * * * * * *.0.0 () μ e n Strong IC GAC* FDGAC* Time(s) Bktrks Time(s) Bktrks Time(s) Bktrks * * * * * * * * 0..0 () μ vr Figure : The time in seons n the numer of ktrks in solving softene ll-intervl series instnes y enforing ifferent onsistenies on the soft lldifferent onstrints with μ e (top) n μ vr (ottom). We give the results for lldifferent with μ e n μ vr in Figure, whih grees well with the theoretil omprison of the three onsistenies. This emonstrtes tht minimum ost flow omputtion is n effiient metho for enforing the onsistenies of projetion-sfe soft glol onstrints. Despite higher omplexity, FDGAC*, the strongest onsisteny oth in terms of pruning n lower oun resoning, is the ler winner ettering strong IC y one to two orers of mgnitue, while GAC* omes in ler seon. In the est se, enforing FDGAC* n remove times more serh noes thn enforing GAC*, n 0 times more thn enforing strong IC. Lst ut not lest, we note tht, without strong IC, Toulr elys the propgtion of n-ry onstrints until only two vriles restrite y the onstrints re not yet ssigne. It is imprtil to solve the enhmrk even with smll vlue of n. Referenes [Ahuj et l., 00] R.K. Ahuj, T.L. Mgnnti, n J.B. Orlin. Network Flows: Theory, Algorithms, n Applitions. Prentie Hll/Person, 00. [Cooper n Shiex, 00] M. Cooper n T. Shiex. Ar Consisteny for Soft Constrints. Artifiil Intelligene, :, 00. [Lrros n Shiex, 00] J. Lrros n T. Shiex. In the Quest of the Best Form of Lol Consisteny for Weighte CSP. In Proeeings of IJCAI 00, pges, 00. [Lrros n Shiex, 00] J. Lrros n T. Shiex. Solving Weighte CSP y Mintining Ar Consisteny. Artifiil Intelligene, (-):, 00. [Mher et l., 00] M. Mher, N. Nroytsk, C.-G. Quimper, n T. Wlsh. Flow-Bse Propgtors for the SE- QUENCE n Relte Glol Constrints. In Proeeings of CP 00, pges, 00. [Petit et l., 00] T. Petit, J.-C. Regin, n C. Bessiere. Speifi Filtering Algorithm for Over-Constrine Prolems. In Proeeings of CP 00, pges, 00. [Regin, 00] J.-C. Regin. Cost-Bse Ar Consisteny for Glol Crinlity Constrints. Constrints, : 0, 00. [Snhez et l., 00] M. Snhez, S. e Givry, n T. Shiex. Menelin Error Detetion in Complex Peigrees using Weighte Constrint Stisftion Tehniques. Constrints, ():0, 00. [Shiex et l., ] T. Shiex, H. Frgier, n G. Verfillie. Vlue Constrint Stisftion Prolems: Hr n Esy Prolems. In Proeeings of IJCAI, pges,. [vn Hoeve et l., 00] W.J. vn Hoeve, G. Pesnt, n L.- M. Rousseu. On Glol Wrming: Flow-se Soft Glol Constrints. J. Heuristis, (-):, 00. [Zytniki et l., 00] Mtthis Zytniki, Christine Gspin, n Thoms Shiex. A New Lol Consisteny for Weighte CSP Deite to Long Domins. In Proeeings of SAC 00, pges, 00. Conlusion Glol onstrints re one of the keys for moeling n solving omplex rel-life prolems. To the est of our knowlege, this is the first suess report of glol onstrints in WCSP solvers with prtil effiieny. Our tehniques mke it possile to enfore generlize versions of existing onsistenies exploiting speifilly hrteristis of WCSPs. Immeite future work inlues stuies of the implementtion of more projetion-sfe soft glol onstrints, fesiility of other forms of onsistenies, experiments on wier vriety of enhmrks. It is lso interesting to investigte if there re other forms of projetion-sfety.