Weighted Constraint Satisfaction Problems with Min-Max Quantifiers

Transcription

1 Weighted Constrint Stisftion Prolems with Min-Mx Quntifiers Jimmy H.M. Lee & Terrene W.K. Mk Deprtment of Computer Siene nd Engineering The Chinese University of Hong Kong Shtin, N.T., Hong Kong Justin Yip Brown University Box 1910, Providene, RI USA Astrt Soft onstrints re funtions returning osts, nd re essentil in modeling over-onstrined nd optimiztion prolems. We re interested in tkling soft onstrined prolems with dversril onditions. Aiming t generlizing the weighted nd quntified onstrint stisftion frmeworks, Quntified Weighted Constrint Stisftion Prolem (QWCSP) onsists of set of finite domin vriles, set of soft onstrints, nd min or mx quntifier ssoited with eh of these vriles. We formlly define QWCSP, nd propose omplete solver whih is sed on lph-et pruning. QWCSPs re useful speil ses of QCOP/QCOP+, nd n e solved s QCOP/QCOP+. Restriting our ttention to only QWCSPs, we show empirilly tht our proposed solving tehniques n etter exploit prolem hrteristis thn those developed for QCOP/QCOP+. Experimentl results onfirm the fesiility nd effiieny of our proposls. Keywords-Constrint Optimiztion, Soft Constrint Stisftion, Quntified Constrint Stisftion I. INTRODUCTION The tsk t hnd is tht of n optimiztion prolem with dversril onditions. As n exmple, we egin with generlized grph oloring prolem in whih numers re used insted of olors. In ddition, the grph is numered y two plyers. The nodes re prtitioned into two sets, A nd B. Plyer 1 will numer set A first, followed y plyer 2 numering set B. The gol of plyer 1 is to mximize the totl differene etween numers of djent nodes, while plyer 2 wishes to minimize the totl differene. The im is to help plyer 1 devising the est strtegy. The exmple is optimiztion in nture, nd the dversries originte from the numers eing pled on nodes y plyer 2. The question n e trnslted to mximizing totl differene for ll possile omintions of numers plyer 2 n write. One wy to solve this prolem is y tkling mny Constrint Optimiztion Prolems [1] or Weighted CSPs [2], where eh of them mximizes the totl differene onditioned on speifi omintion of numers given y plyer 2. Solving these su-prolems We thnk the nonymous referees for their onstrutive omments. The work desried in this pper ws sustntilly supported y grnts (CUHK nd CUHK413808) from the Reserh Grnts Counil of Hong Kong SAR. We lso thnk In Gent for giving us QCSP-Solve, whih serves s sis for our solver. independently, however, defies opportunities for exploiting glol prolem struture nd hrteristis, nd reuse of omputtion. Another wy is to model the prolem s QCSP [3] y finding whether there exists omintions of numers for plyer 1 for ll numer plements y plyer 2 suh tht the totl differene is less thn ost k. Trying different vlues of k progressively in seprte QCSPs flls short in utilizing the ojetive funtion to guide serh glolly. We propose to omine the est of oth worlds. Weighted CSPs re minimiztion in nture. We introdue the mx quntifier to further llow min-mx opertions on onstrint osts in Weighted CSPs. A Quntified Weighted Constrint Stisftion Prolem (QWCSP) onsists of n ordered sequene of finite domin vriles, set of soft onstrints, nd min or mx quntifier ssoited with eh vrile. Note tht the existentil ( ) nd universl ( ) quntifiers in QCSPs n e expressed using min nd mx respetively. WCSPs nd QCSPs re thus lso speil ses of QWCSPs. We define solutions of QWCSP, nd show how rnh-nd-ound serh with lph-et pruning n e pplied to solve QWCSP. Generl pruning onditions of lph-et serh re defined nd disussed. The new frmework ims to nswer suh interesting questions s Wht is the est pln I n hoose to minimize ll the possile penlties for the worst se senrio? QWCSPs n e modeled s QCOPs [4], whih re more generl frmework. We give onstrution method followed y n exmple. We perform experimentl evlutions, ompring our proposed solving tehniques nd those for QCOPs, on three enhmrks to show the effiieny nd fesiility of our frmework. II. BACKGROUND We give the si definitions of WCSPs nd QCSPs. A weighted onstrint stisftion prolem [2] (WCSP) is tuple (X, D, C,k), wherex = {x 1,...,x n } is finite set of vriles nd D = {D 1,...,D n } is set of domins of possile vlues. We denote x i = v i e n ssignment ssigning vlue v i D i to vrile x i, nd the set of ssignments l = {x 1 = v 1,x 2 = v 2,...,x n = v n } e omplete ssignment on vriles X,wherev i is the vlue ssigned to x i.aprtil ssignment l[s] is projetion of l

2 onto vriles in S X. C is set of (soft) onstrints, eh C S of whih represents funtion mpping tuples orresponding to ssignments on suset of vriles S, toost vlution struture V (k) = ([0...k],, ). The struture V (k) ontins set of integers [0...k] with stndrd integer ordering. Addition is defined y = min(k, + ). For ny integer nd where, sutrtion is defined y = if k, nd = k if = k. Without loss of generlity, we ssume the existene of C denoting the lower ound of the minimum ost of the prolem. If it is not defined, we ssume C =0.Theost of omplete ssignment l in X is defined s: ost(l) =C C s C C s (l[s]) A omplete ssignment l on X is fesile if ost(l) <k, nd is solution of WCSP if l hs the minimum ost mong ll fesile tuples. A quntified onstrint stisftion prolem [3] (QCSP) P is tuple (X, D, C, Q), wherex =(x 1,x 2,...,x n ) is n ordered sequene of vriles, D =(D 1,D 2...,D n ) is n ordered sequene of finite domins, C = C 1 C 2... C e is onjuntion of onstrints, nd Q = (Q 1,...,Q n ) is quntifier sequene in whih eh Q i is either (existentil, there exists ) or (universl, for ll ) nd ssoited with x i. A onstrint C k C onsists of sequene X k =(x k1,...,x kr ) of r>0vriles s.t. X k is susequene of X nd D k is susequene of D. C k hs n ssoited set A[C k ] D k1... D kr of tuples whih speify llowed omintions of vlues for the vriles in X k.letfirstx(p) returns the first unssigned vrile in the vrile sequene. If there re no suh vriles, it returns. Thesemntis of QCSP P is defined reursively s follows: (1) In se firstx(p) =, if ll onstrints C k C re stisfile, P is stisfile; nd if ny onstrint fils, P is unstisfile. (2) Otherwise, let firstx(p) = x i.ifq i = then P is stisfile iff there exists vlue D i suh tht the simplified prolem P with ssigned to x i is stisfile. If Q i = then P is stisfile iff for ll vlues D i the simplified prolem P with ssigned to x i is stisfile. III. QUANTIFIED WEIGHTED CSPS Stndrd WCSPs re minimiztion in nture, we im t lso optimizing prolems with dversril onditions, y modeling dversries using mx quntifiers. A Quntified Weighted Constrint Stisftion Prolem (QWCSP) P is tuple (X, D, C, Q,k), wherex =(x 1,...,x n ) is defined s n ordered sequene of vriles, D = (D 1,...,D n ) is n ordered sequene of finite domins, C is set of soft onstrints s in WCSPs, Q =(Q 1,...,Q n ) is quntifiers sequene where Q i is either mx or min ssoited with x i, nd k is the glol upper ound. In QWCSP, ordering of vriles is importnt. Without loss of generlity, we ssume vriles re ordered y their indies. We define vrile with min (mx resp.) quntifier to e minimiztion vrile (mximiztion vrile resp.). Let P[x i1 = i1 ][x i2 = i2 ]...[x im = im ] e the suprolem otined from P y ssigning vlue i1 to vrile x ii, ssigning vlue i2 to vrile x i2,..., ssigning vlue im to vrile x im. Suppose l is omplete ssignment of P. We reuse the definition of firstx(p) definedinqcsps. The A-ost(P) of QWCSP P is defined reursively s follows: ost(l), if firstx(p) = A-ost(P) = mx(m i), if firstx(p) =x i nd Q i =mx min(m i), if firstx(p) =x i nd Q i =min where M i = {A-ost(P[x i = v]) v D i }. A QWCSP P is stisfile iff A-ost(P) < k. Similr to QCSPs, we define lok of vriles in QWCSP P to e mximl susequene of vriles in X whih hs the sme quntifiers. Chnging the vrile ordering within the sme lok of vriles does not hnge the A-ost of QWCSP. Exmple 1: Given QWCSP P with the ordered sequene of vriles (x 1,x 2,x 3 ), domins D 1 = {,, },D 2 = {, }, nd D 3 = {,, }, the set of onstrints represented in Figure 1, the quntifier sequene (Q 1 =mx,q 2 =min,q 3 =mx), nd the glol upper ound k. The prolem is to find the A-ost of P. Figure 1 indites there re 3 unry onstrints C 1,C 2,C 3 nd 2 inry onstrints C 1,2,C 2,3. For unry onstrints, non-zero unry osts re depited inside irle nd domin vlues re pled ove the irle. For inry onstrints, non-zero inry osts re depited s lels on edges onneting the orresponding pir of vlues. Only non-zero osts re shown. We show the omputtion for the A-ost of the QWCSP P s follows: A-ost(P) = mx { min { mx {A-ost(P[x 1 = v 1][x 2 = v 2][x 3 = v 3])}}} v 1 D 1 v 2 D 2 v 3 D 3 = mx{ min{mx{ost(,, ),ost(,, ),ost(,, )},...}, min{mx{ost(,, ),ost(,, ),ost(,, )},...}, min{mx{ost(,, ),ost(,, ), ost(,, )},...}} = mx{ min{mx{10, 5, 4}, mx{11, 8, 6}}, min{mx{7, 2, 1}, mx{7, 4, 2}}, min{mx{6, 1, 0}, mx{8, 5, 3}}} = mx{ 10, 7, 6} =10 If k>10 in Exmple 1, then the prolem is stisfile. Otherwise, Exmple 1 is unstisfile. Solution of WCSP is omplete ssignment with the minimum osts, while solution in QCSPs is winning strtegy [5]. In this pper, we define solution of QWCSP P s omplete ssignment {x 1 = v 1,...,x n = v n } s.t.: A-ost(P) =A-ost(P[x 1 = v 1 ]...[x i = v i ]), 1 i n. Extrting solutions is esy fter omputing the A-ost of QWCSP.

3 10 x 1 x 2 x 3 mx x min x mx x 3 Figure 1. Constrints for Exmple 1 Figure 2 shows n dpttion of the stndrd leling tree [1] for Exmple 1. The A-ost for eh su-prolem is pled inside the orresponding node representing the su-prolem. Both WCSPs nd QCSPs re speil ses of QWCSPs. We stte without proof the following theorems. Theorem 1: A WCSP P = (X, D, C,k) n e trnsformed y Krp redution [6] to n equivlent QWCSP P =(X, D, C, Q,k) with ll Q i Qequl to the min quntifier. Given hrd onstrint C. We n onstrut soft onstrint C on the sme set of vriles. A soft onstrint C returns ost 0 if C is stisfile on the sme set of ssignments; otherwise, C returns ost k. Theorem 2: AQCSPP = X, D, C, Q n e trnsformed y Krp redution [6] to n equivlent QWCSP P = (X, D, C, Q, 1) where C is the set of onstrints onstruted from C. For eh Q i Q,ifx i Xis n existentil vrile in QCSP, then Q i is min quntifier; otherwise, Q i is mx quntifier. Corollry 3: QCSPs nd WCSPs re speil ses of QWCSPs, whih re PSPACE-hrd. IV. BRANCH &BOUND WITH CONSISTENCIES This setion outlines omplete solver for QWCSPs. The key ide of the solver is tht, y pplying lph-et pruning [7] in the Brnh & Bound serh nd dpting onsisteny tehniques used in WCSP [2], we n estimte the A-ost erly in the serh so s to redue the serh spe. We first disuss lph-et pruning. Then we disuss generl onditions whih n led to prunings nd ktrkings. Due to spe limittions, we skip detiled node nd r onsisteny notions whih were modified nd integrted with lph-et pruning to solve QWCSPs more effiiently. A. Alph-Bet Pruning Alph-et pruning ttempts to redue serh nodes in minimx lgorithm y exploiting () semntis of the mx nd min quntifiers nd () the upper nd lower ounds of the osts of previously visited nodes. We n pply lph-et pruning in the Brnh & Bound serh diretly in solving QWCSPs s only mx nd min quntifiers re mx x 1 min x 2 mx x Figure 2. Leling Tree for Exmple X 6 X X X X X X X X X Figure 3. Leling Tree for Exmple 1 fter pplying lph-et pruning llowed. Alph-et pruning is lso used for solving reltime online QCSPs [8]. Algorithm 1 is high-level strtion of QWCSP solver. We first neglet the propgtion routine in lines 7 31 nd disuss the si lph-et pruning lgorithm. The serh strts with lph_et(p, 0,k). The ounds l nd u re the rnge of osts found y the lph-et pruning lgorithm. QWCSP propgtors further exploit these ounds to hieve stronger propgtion. We use P[x j u] to denote funtion pruning vlue u of vrile x j in P, nd P[x i = v] to denote funtion ssigning vlue v to vrile x i in P. Both funtions return the modified prolem. Line 3 is the se se in whih ll vriles re ound. The routine ost returns the ost of the omplete ssignment. Lines give the min routine of the trditionl lphet pruning lgorithm. We only explin the ost for the min quntifier, sine tht of mx is similr. The for loop in lines 6 42 evlutes ll su-prolems P[x i = v] y reursively invoking the lph-et lgorithm. Sine the gol is to find minimum vlue, the upper ound is updted. When the upper ound is less thn the lower ound (line 38), it triggers the short-ut to rek out of the remining serh sine every vlue returned y susequent lls will e dominted y the urrent ounds. The funtion lph_et ends y returning the upper ound for the min quntifier (line 41). We illustrte the ode with n exmple. Exmple 2: Consider the tree in Figure 2, nd ssume vlues re leled in the sequene of [,, ]. The serh strts with lph_et(p, 0,k). Consider the node P = P[x 1 = ], whih first visits its su-prolem P [x 2 = ] y

4 Algorithm 1 A QWCSP Solver 1: funtion lph_et(p,l,u): 2: if firstx(p) == then 3: return ost(p) 4: end if 5: x i = firstx(p) 6: for v D i do 7: {Pruning using QWCSP semntis} 8: hnged = true 9: while hnged do 10: hnged = flse 11: for j [i..n] do 12: for u D j do 13: p l = pprox_l(p,x j = u) 14: if u <= p l then 15: if Q j == min then 16: P = P[x j u],hnged = true 17: else 18: return u 19: end if 20: end if 21: p u = pprox_u(p,x j = u) 22: if p u <= l then 23: if Q j == min then 24: return l 25: else 26: P = P[x j u],hnged = true 27: end if 28: end if 29: end for 30: end for 31: end while 32: {Bsi lph-et pruning} 33: if Q i == min then 34: u = min(u, lph_et(p[x i = v],l,u)) 35: else 36: l = mx(l, lph_et(p[x i = v],l,u)) 37: end if 38: if u <= l then 39: rek 40: end if 41: return (Q i == min)?u : l 42: end for lling lph_et(p [x 2 = ], 0,k) nd vlue of 10 is returned. Sine Q 2 =min, the upper ound is updted. The routine then invoke lph_et(p [x 2 = ], 0, 10). After visiting P [x 2 = ][x 3 = ], we get n A-ost of 11 for the su-prolem. The quntifier here is mx, theost is greter thn the upper ound nd the ondition in line 38 holds. No mtter wht osts the remining su-prolems produe, they hve no impt on the solution nd short-ut to rek out of the remining serh is triggered. Hene, the su-prolems P [x 2 = ][x 3 = ] nd P [x 2 = ][x 3 = ] re not explored. Figure 3 illustrtes the nodes pruned, denoted y the symol X, y lph-et pruning. B. Consisteny Tehniques In trditionl CSPs, we enfore different levels of onsisteny to prune infesile domin vlues nd hene redue the serh spe. In WCSPs, the onsisteny lgorithms tke the ost of onstrints into ount. Vrious onsisteny notions (e.g. NC*, AC* [2], FDAC*, EDAC*, OSAC, nd VAC [9]) hve een proposed nd proven to e useful in improving solver performne. Suh tehniques, however, nnot e diretly pplied to QWCSP sine the quntifiers hnge the semntis of onstrints. In prtiulr, pplying these onsisteny notions on onstrints overing on mx vriles my result in unsound prunings. Consisteny notions for QWCSPs need to tke quntifiers into ount. To prune vlues of QWCSP, the min ide is tht if the A-ost of su-prolem P = P[x i = v] is greter thn or equl to the upper ound u (less thn or equl to the lower ound l resp.), the pruning tehniques in lph-et n e pplied. Let P[x 1..i 1 = v 1..i 1,x i = v] denote the suprolem P[x 1 = v 1 ][x 2 = v 2 ]...[x i 1 = v i 1 ][x i = v]. Formlly, we onsider two onditions: v D i s.t. v 1 D 1,..., v i 1 D i 1 : A-ost(P[x 1..i 1 = v 1..i 1,x i = v]) u (1) A-ost(P[x 1..i 1 = v 1..i 1,x i = v]) l (2) When ny of the ove onditions is stisfied, we n pply lph-et pruning ording to Tle I. Tle I WHEN CAN WE PRUNE/BACKTRACK A-ost u l Q i =min prune v ktrk Q i =mx ktrk prune v Theorem 4: Given QWCSP P. If Condition (1)/(2) for P is stisfied, pplying prunings nd ktrkings ording to Tle I is sound. Proof: (Sketh) Resons to perform prunings nd ktrking for min nd mx re symmetril. We only desrie the se where Q i =min. Suppose Condition (1) holds. We onsider A-ost(s) for su-prolems P[x 1..i 1 = v 1..i 1 ]. Without loss of generlity, we write P i 1 to e one of these su-prolems P[x 1..i 1 = v 1..i 1 ] y fixing vlues v 1 D 1, v 2 D 2,..., v i 1 D i 1. We will see the proof using P i 1 pplies for ll su-prolems P[x 1..i 1 = v 1..i 1 ], regrdless on whih vlues we fix. Given Q i =min,we otin: A-ost(P i 1 )= minp i 1 [x i = ] D i If A-ost(P i 1 ) <u, the following must e true: v D i where v v s.t. A-ost(P i 1 [x i = v ]) <u Pruning vlue v does not hnge the A-ost of P i 1.If A-ost(P i 1 ) u, i.e.p i 1 must not led to solutions, the following must e true: v D i, A-ost(P i 1 [x i = v ]) u After pruning vlue v, either domin wipe out ours or A-ost(P i 1 ) u. For oth ses, the su-prolem P i 1 nnot led to solutions. Comining the two ses, pruning vlue v does not hnge the prolem from unstisfile to stisfile(,nd vie vers). We now disuss Condition (2). Similr to the previous se, we onsider the A-ost for these su-prolems

5 2 Upper ound u : 10 Lower ound l : 1 0 Upper ound u : 10 Lower ound l : 1 mx x1 mx x min x 2 mx x mx x 3 min x Figure 4. Leling Tree for Exmple 3 P[x 1..i 1 = v 1..i 1 ],ndwefixp i 1 to e one of these su-prolems similrly. Given Q i =min, we otin: A-ost(P i 1 )= minp i 1 [x i = ] D i By Condition (2), P i 1 [x i = v] l holds, nd therefore: A-ost(P i 1 ) l Rell A-ost(P[x 1..i 1 = v 1..i 1,x i = v]) l pplies regrdless on whih vlue v 1, v 2,..., v i 1 we fix. Therefore, we otin: v 1 D 1,..., v i 1 D i 1, A-ost(P[x 1..i 1 = v 1..i 1]) l We n esily otin the following results using the definition of A-ost for QWCSP: A-ost(P) l. QWCSP P must e unstisfile, nd the solver n ktrk. Exmple 3: Given QWCSP P with the ordered sequene of vriles (x 1,x 2,x 3 ), domins D 1 = {,, },D 2 = {, }, ndd 3 = {,, }, the quntifier sequene (Q 1 = mx,q 2 = min,q 3 = mx),ndthe glol upper ound 10. Suppose the A-ost for su-prolem P[x 1 = ] is 1. Figure 4 shows the upper ound u, lower ound l, ndthea-osts for the remining su-prolems. By inspeting the figure, we n esily oserve Condition (2) holds: D 3 s.t. v 1 D 1, v 2 D 2, A-ost(P[x 1 = v 1 ][x 2 = v 2 ][x 3 = ]) l By Tle I, we n prune vlue of x 3. We n esily oserve the solution must not ontin the ssignment x 3 =, nd therefore, we n prune the vlue. After pruning vlue of x 3, we n esily oserve Condition (1) holds: D 2 s.t. v 1 D 1, A-ost(P[x 1 = v 1 ][x 2 = ]) u Similrly, we n prune vlue of x 2. Exmple 4: Suppose the quntifier sequene of Exmple 3 is repled y (Q 1 =mx,q 2 =mx,q 3 = min), nd the A-ost for su-prolem P[x 1 = ] remins unhnged (A-ost(P[x 1 = ]) = 1). Figure 5 shows the upper ound u, lower ound l,ndthea-osts for the remining su-prolems. Costs for eh omplete ssignment remin Figure 5. Leling Tree for Exmple 4 the sme s in Exmple 3. The only differene is the modified A-osts for su-prolems, resulting from the hnge in quntifiers. By inspeting the figure, we n esily oserve Condition (2) still holds: D 3 s.t. v 1 D 1, v 2 D 2, A-ost(P[x 1 = v 1 ][x 2 = v 2 ][x 3 = ]) l As Q 3 =min, we n esily oserve ll the A-osts for su-prolems P[x 1 = v 1 ][x 2 = v 2 ], v 1 D 1,v 2 D 2 must e less thn or equl to the l. By indution, we n onlude su-prolems P[x 1 = v 1 ], v 1 D 1 must e less thn or equl to the l, nd finlly otin A-ost(P) l. Therefore following Tle I, the solver n ktrk. One wy to hek Condition (1)/(2) is to find the ext vlue of the A-ost for eh su-prolem, whih is omputtionlly expensive. The prolem is essentilly equivlent to determining if vrile ssignment is solution of lssil CSPs in generl, whih is NP-hrd. A ommon tehnique in onstrint progrmming is to formulte onsisteny notions nd lgorithms, whih im t extrting nd mking informtion in prolem expliit. Useful informtion inludes pruning nd ost informtion. Here we pply the sme ide. We use effiient wys to extrt good upper ound nd lower ound of A-ost, so s to ktrk or identify non-solution vlues from domins erly in the serh. In the QWCSP solver (Algorithm 1), lines 7 31 prune or ktrk ording to the onditions speified in Tle I. Sine finding A-ost is diffiult, the lgorithm finds the pproximted ounds (pprox_l in line 13, nd pprox_u in line 21). Funtions pprox_l(p,x i = v) nd pprox_u(p,x i = v) find the pproximte A-ost for the set S of su-prolems, where: S = {P[x 1..i 1 = v 1..i 1,x i = v] v 1 D 1,...,v i 1 D i 1} suh tht: P S, A-ost(P ) pprox_u(p,x i = v),nd P S, A-ost(P ) pprox_l(p,x i = v)

6 pprox_u(p,x i = v) is tight if: mx A-ost(P )=pprox_u(p,x i = v) P S Similrly, pprox_l(p,x i = v) is tight if: min A-ost(P )=pprox_l(p,x i = v) P S Corollry 5: Given QWCSP P, If pprox_u(p,x i = v) l, we n prune vlue v of vrile x i if Q i = mx, nd perform ktrk if Q i =min.ifpprox_l(p,x i = v) u, we n prune vlue v of vrile x i if Q i =min, nd perform ktrk if Q i =mx. Proof: (Sketh) We n esily oserve the following: l pprox_u(p,x i = v) P[x 1..i 1 = v 1..i 1,x i = v] u pprox_l(p,x i = v) P[x 1..i 1 = v 1..i 1,x i = v] v 1 D 1,v 2 D 2,...,v i 1 D i 1. Condition pprox_u(p,x i = v) l implies Condition (2), nd ondition pprox_l(p,x i = v) u implies Condition (1). We then pply Tle I ording to these onditions. We hve designed two kinds of onsisteny notions: node onsisteny nd r onsisteny, whih extrt the pproximted upper nd lower ounds (hene implement pprox_u nd pprox_l) for QWCSPs. Unfortuntely, we hve to skip the tehnil detils due to spe limittions. V. QCOP VERSUS QWCSP QWCSPs re speil ses of QCOP [4]. Given QWCSP, this setion shows how to onstrut QCOP instne y the Soft As Hrd pproh [10], whih ws used to onstrut lssil COP instnes from WCSP instnes. In the next setion, we provide empiril evidene to demonstrte tht our proposed solving tehniques re more effiient thn those for QCOPs for tkling QWCSPs. A. Definitions of QCOP We first give the definition of QCOP [4]. A restrited quntified set of vriles (rqset) is tuple (q, W, C) where q {, }, W is suset of vriles W V,ndC is CSP. A prefix P of rqsets is sequene of rqsets ((q 1,W 1,C 1 ),...,(q n,w n,c n )) suh tht W i W j =, i j. We define rnge(p ) for prefix P with n rqsets to e [1..n], efore i (P ) to e j i W j,ndnu i (P )= min j>i {j q j = } to e the index of the next universl lok of vriles loted fter n index i. If no suh index exists, we denote nu i (P ) y n +1.LetA e set of ggregte nmes nd F set of ggregte funtions. An ggregte is n tom of the form : f(x) where A, f F,nd X V A.Anoptimiztion ondition is n tom of the form min(x), mx(x) where X V A or the tom ny. An tom min(x) (mx(x) resp.) mens the user is interested in strtegies tht minimize (mximize resp.) this vlue, while ny indites the user does not re out the returned strtegy. A -orqset is tuple (,W,C,o) where (,W,C) is rqset nd o is n optimiztion ondition. A -orqset is tuple (,W,C,A) where (,W,C) is rqset nd A is set of ggregtes. We denote nmes of the set of ggregtes A in -orqset y nmes(a). Anorqset is either -orqset or -orqset. A QCOP+ is pir (P, G) where G is CSP nd P =(orq 1,...,orq n ) is prefix of orqsets suh tht i rnge(p ), with k = nu i (P ): 1)iforq i =(,W,C,o) with o = min(x) or o = mx(x), then we must hve X efore k 1 (P ) (k <n+1?nmes(a k ): ), nd 2) if orq i =(,W,C,A), then for ll : f(x) in A, wemust hve X efore k 1 (P ) (k <n+1?nmes(a k ): ). A QCOP is QCOP+ in whih no orqset hs restritions, i.e. no onstrints in the CSP of ll orqsets. B. Trnsforming QWCSPs into QCOPs We n trnsform ny QWCSP P =(X, D, C, Q,k) into QCOPP = (P,G ) sed on the modified Soft As Hrd pproh s follows. For eh vrile x i in P, there is n orqset orq i =(, {x i },C i,o i ) in P,whereC i hs no onstrints. For every soft onstrint C in C, thereis orresponding ost vrile x in the CSP G with domin eing equl to ll possile osts given y C. We onstrut n uxiliry ost vrile s in CSP G whih is equl to the sum of ll ost vriles x. A onstrint s<kis dded to restrit the totl ost to e less thn the glol upper ound. If x i is minimiztion vrile, then we dd o i = min(s); otherwise, o i =mx(s). Suppose C on set S of vriles giving ost m when tuple of ssignment l is ssigned. There is reified onstrint in the CSP G restriting x to tke vlue m if vriles in S re ssigned with tuple l. Exmple 5: Given QWCSP P with n ordered sequene of vriles (x 1,x 2 ), domins D 1 = D 2 = {, }, set of onstrints {C 1,C 2 }, quntifier sequene {Q 1 = min,q 2 = mx), nd glol upper ound k = 7. C 1 () =0,C 1 () =5,C 2 () =1,C 2 () =3. The QCOP P n e expressed using the Soft As Hrd pproh s showninfigure6. x 1 D 1 x 2 D 2 min s mx s x C 1 {0, 5},x C 2 {1, 3},s {1, 3, 6} s = x C 1 x C 2 s<7 [x 1 = ] [x C 1 =0] [x 1 = ] [x C 1 =5] [x 2 = ] [x C 2 =1] [x 2 = ] [x C 2 =3] Figure 6. Trnsformed QWCSP in Exmple 5 Theorem 6: A QWCSP P n e trnsformed into QCOP P.TheA-ost of P n e found y solving the optiml strtegy [4] of P.

7 The proof follows diretly from the Soft As Hrd onstrution. In prtiulr, A-ost of P is equl to the vlue ssigned to s in the optiml strtegy [4] of P. The entrl theme of this pper is to show QWCSP, more restrited ut useful sulss of QCOP+, n e solved more effiiently. Bsed on WCSPs, QWCSP s onsisteny tehniques redistriute onstrint osts y projetions nd extensions [2]. In turn, we onjeture tht this llows extr prunings over the lssil onsistenies used in QCOP+. The sitution is similr to how Lee nd Leung [11] show WCSP (soft pproh) onsistenies to e stronger thn lssil optimiztion used in Soft As Hrd. VI. PERFORMANCE EVALUATION In this setion, we ompre the QCOP+ solver QeCode ginst our solver in three progressive modes: Alph-et pruning, Node Consisteny (NC), nd Ar Consisteny (AC). Vlues re leled in stti lexiogrphi order. We generte 20 instnes for eh enhmrk s prtiulr prmeter setting. Results for eh enhmrk re tulted with numer of solved instnes, verge time used, nd verge numer of tree nodes enountered. We tke verge for solved instnes only. Winning entries for verge time used nd verge numer of tree nodes enountered re highlighted in old. A symol - represents ll instnes fil to run within the time limit of 900 seonds. The experiment is onduted on Pentium 4 3.2GHz with 3GB memory. We ompre our solver ginst QeCode, whih uses minimx. All QWCSP instnes re trnsformed to QCOP using the Soft As Hrd pproh outlined. We note lph-et prunings n e employed for QCOP+, ut we elieve there will e less prunings y enforing lssil onsistenies. A. Rndom Generted Prolems We generte rndom inry QWCSPs with prmeters (n, s, d), where n is the numer of vriles, s is the domin size for eh vrile, nd d is the proility for inry onstrint to our etween two vriles. We purposefully do not generte unry onstrints to mke the prolem hrder to solve. The osts for eh inry onstrint re generted uniformly in [0...30]. Quntifiers re generted rndomly with hlf proility for min (mx resp.), nd numer of quntifier levels vry from instnes to instnes. Tle II shows the results. For ll instnes, even just lph-et pruning is two orders of mgnitude fster thn QeCode, whih nnot hndle even modertely sized instnes. NC nd AC oth run fster thn lph-et pruning, with the serh spe drmtilly deresed nd runtime signifintly fster. AC, utilizing informtion from oth unry nd inry onstrints, performs est mong ll solver modes. B. Grph Coloring Gme We hve generted instnes (v,, d) for grph oloring gme similr to the one in the introdution, where v is n even numer of nodes in the grph, is the rnge of numers llowed to ple, nd d is the proility of n edge etween two verties. Plyer 1 (Plyer 2 resp.) is ssigned to ply the odd (even resp.) numered turns, plyers ply in turn, nd in eh turn node is numered. The node orresponding to eh turn is generted rndomly. Tle III shows the results. Similr to Rndom Prolems, lph-et pruning runs fster thn QeCode in ll instnes two orders of mgnitude fster. NC nd AC oth run fster thn lphet pruning. Compring AC nd lph-et pruning, we n gin up to six times speedup for AC, whih prunes up to two-third of the serh spe of NC. Agin, AC etters in lmost ll instnes in runtime. C. Min-Mx Resoure Allotion Prolem Suppose N units of resoures re lloted to t tivities. We let x i e the mount of resoures lloted to tivity i, nd funtion i (x i ) returns the ost inurred from tivity i y lloting x i units of resoures the tivity. The resoure llotion prolem [12] is to find n optiml resoure llotion so s to minimize the totl ost. Suppose now there re s funtions s i. The min-mx resoure llotion prolem [13] is to find resoure llotion to minimize the mximum ost funtions. Tle IV shows the results. Alph-et pruning runs n order of mgnitude fster thn QeCode. Agin, NC nd AC oth run fster thn lph-et pruning. AC n prune up to 90% of the serh spe of NC nd runs the fstest in ll instnes. VII. CONCLUDING REMARKS We define the QWCSP frmework for modeling optimiztion prolems with dversries. Our work llows us to model nd solve interesting prolems, suh s grph oloring gme nd min-mx resoure llotion prolem, effiiently. We propose nd implement omplete solver inorporting lph-et pruning into rnh-nd-ound. We evlute our proposl using three enhmrks nd ompre with QeCode. Experimentl results show onsisteny enforement is worthwhile in generl nd our solver is orders of mgnitude fster thn QeCode in tkling QWCSPs, lthough QeCode is more generl solver for solving QCOP/QCOP+. When restriting ttention to QWCSPs, we re le to devise speifi inferene to prune nd guide serh. Our results suggest lph-et pruning tehniques n e future possile diretion for enhning QeCode. Brown et. l propose similr frmework, Adversril CSPs [14], whih fouses on the se where two opponents tke turns to ssign vriles, eh trying to diret the solution towrds their own ojetives. They desrie the proess s gme-tree serh, nd the solving lgorithm is sed on the minimx lgorithm. Our work pushes the ide further y: 1) pplying lph-et pruning, nd 2) exploiting osts informtion using soft onstrint tehniques. Another relted work is Stohsti CSPs [15], whih n represent

8 Tle II RANDOM GENERATED PROBLEM QeCode Alph-et Node Consisteny Ar Consisteny (n, s, d) #solve Time #nodes #solve Time #nodes #solve Time #nodes #solve Time #nodes (9, 5, 0.4) ,394, , , ,221 (9, 5, 0.6) ,394, , , ,024 (12, 5, 0.4) ,967, , ,866 (12, 5, 0.6) ,782, , ,710 (16, 5, 0.4) ,269, ,780, ,657,203 (16, 5, 0.6) ,859, ,085, ,909,388 (18, 5, 0.4) ,210, ,514,426 (18, 5, 0.6) ,171, ,613,430 Tle III GRAPH COLORING GAME QeCode Alph-et Node Consisteny Ar Consisteny (v,, d) #solve Time #nodes #solve Time #nodes #solve Time #nodes #solve Time #nodes (10, 4, 0.4) ,446, , , ,580 (10, 4, 0.6) ,446, , , ,617 (12, 4, 0.4) , , ,124 (12, 4, 0.6) , , ,088 (16, 4, 0.4) ,050, ,002, ,523 (16, 4, 0.6) ,213, , ,691 (18, 4, 0.4) ,095, ,315,212 (18, 4, 0.6) ,295, ,711,880 Tle IV MIN-MAX RESOURCE ALLOCATION QeCode Alph-et Node Consisteny Ar Consisteny (t, N, s) #solve Time #nodes #solve Time #nodes #solve Time #nodes #solve Time #nodes (10, 8, 7) , , , ,317 (12, 8, 7) ,889, , , ,096 (12, 9, 7) ,408, , , ,563 (13, 9, 7) ,461, ,732, , ,831 (13, 9, 8) ,455, ,700, , ,526 dversries y known proility distriutions. Proilities for every tion deided y dversries re known eforehnd. We then seek tions to minimize/mximize the expeted ost for ll possile senrios. Our work is similr in the sense tht we re minimizing the ost for the worst se senrio. It will e interesting to study how to model non-zero sum optimiztion prolems, llow restrited quntiers [16] for soft onstrints, devise r onsisteny lgorithms for high rity soft onstrints, nd investigte on vrile / vlue ordering heuristis. REFERENCES [1] K. Apt, Priniples of Constrint Progrmming. New York, USA: Cmridge University Press, [2] J. Lrros nd T. Shiex, Solving weighted CSP y mintining r onsisteny, Artifiil Intelligene, vol. 159, no. 1-2, pp. 1 26, [3] L. Bordeux nd E. Monfroy, Beyond NP: Ar-onsisteny for quntified onstrints, in CP 02, 2002, pp [4] M. Benedetti, A. Lllouet, nd J. Vutrd, Quntified onstrint optimiztion, in CP 08, 2008, pp [5] L. Bordeux, M. Cdoli, nd T. Mnini, CSP properties for quntified onstrints: Definitions nd omplexity, in AAAI 05, 2005, pp [6] S. Aror nd B. Brk, Computtionl Complexity: A Modern Approh, 1st ed. Cmridge University Press, [7] S. J. Russell nd P. Norvig, Artifiil Intelligene: A Modern Approh. Person Edution, [8] D. Stynes nd K. N. Brown, Reltime online solving of quntified CSPs, in CP 09, 2009, pp [9] M. Cooper, S. de Givry, M. Snhez, T. Shiex, M. Zytniki, nd T. Werner, Soft r onsisteny revisited, Artifiil Intelligene, vol. 174, no. 7-8, pp , [10] T. Petit, J.-C. Régin, nd C. Bessière, Speifi filtering lgorithms for over-onstrined prolems, in CP 01, 2001, pp [11] J. H. M. Lee nd K. L. Leung, Towrds effiient onsisteny enforement for glol onstrints in weighted onstrint stisftion, in IJCAI 09, 2009, pp [12] K. M. Mjelde, Methods of Allotion of Limited Resoures. John Wiley, [13] G. Yu, Min-mx optimiztion of severl lssil disrete optimiztion prolems, Journl of Optimiztion Theory nd Applitions, vol. 98, pp , [14] K. N. Brown, J. Little, P. J. Creed, nd E. C. Freuder, Adversril onstrint stisftion y gme-tree serh, in ECAI 04, 2004, pp [15] T. Wlsh, Stohsti onstrint progrmming, in ECAI 02, 2002, pp [16] M. Benedetti, A. Lllouet, nd J. Vutrd, QCSP mde prtil y virtue of restrited quntifition, in IJCAI 07, 2007, pp