BnB-ADOPT + with Several Soft Arc Consistency Levels

Transcription

1 BnB-ADOPT + wth Severl Soft Arc Consstency Levels Ptrc Guterrez nd Pedro Meseguer Astrct. Dstruted constrnt optmzton prolems cn e solved y BnB-ADOPT +, dstruted synchronous serch lgorthm. In the centrlzed cse, locl consstency technques ppled to constrnt optmzton hve een shown very enefcl to ncrese performnce. In ths pper, we comne BnB-ADOPT + wth dfferent levels of soft rc consstency, propgtng uncondtonl deletons cused y ether the enforced locl consstency or y dstruted serch. The new lgorthm mntns BnB-ADOPT + optmlty nd termnton. In prctce, ths pproch decreses sustntlly BnB- ADOPT + requrements n communcton cost nd computton effort when solvng commonly used enchmrks. INTRODUCTION There s n ncresng nterest n solvng constrnt optmzton prolems (COP) n dstruted form. Often t occurs tht dfferent prolem elements re dstruted mong utonomous gents, nd they cnnot e grouped nto sngle gent for prvcy or for other resons (for exmple, consder dstruted meetng schedulng [7] or sensor networks pplctons []). In ths cse, we tlk out dstruted COP (DCOP). To solve them, dstruted lgorthms re needed, to cheve n optml soluton wthout onng ll prolem elements nto sngle gent. Snce they re sed on messge pssng, communcton costs hve to e ncluded when evlutng them. ADOPT [6] s n synchronous dstruted serch lgorthm for DCOP solvng. It hs een mproved n BnB-ADOPT [8], whch chnged the orgnl est-frst strtegy for depth-frst, otnng etter performnce. Ths lgorthm hs lso een mproved removng some redundnt messges n BnB-ADOPT + [], whch s currently one of the most performnt synchronous dstruted serch lgorthms for DCOP solvng to optmlty. In the centrlzed cse, COPs re often formulted usng soft constrns [5]. The stndrd serch solvng lgorthm s rnch-ndound (BnB). Mntnng some locl consstency on soft constrnts durng BnB serch cuses sustntl mprovements n performnce [4, ]. Tkng nsprton from ths fct, we hve explored locl consstency mntennce of soft constrnts when solvng DCOPs. Notce tht locl consstences re conceptully equl n the centrlzed/dstruted cses. However, mntnng locl consstences durng dstruted serch requres dfferent technques thn n the centrlzed cse, where ll prolem elements re vlle to the sngle gent performng the serch. Mntnng locl consstences keeps the optmlty nd termnton of synchronous dstruted serch. Specfclly, we hve tken BnB-ADOPT + s synchronous dstruted serch lgorthm, on top of whch we mntn AC nd FDAC versons of soft rc consstency. Then, we present the new lgorthms BnB-ADOPT + -AC nd BnB-ADOPT + -FDAC. They IIIA, CSIC, Cmpus UAB, 089 Bellterr, Spn. {ptrc pedro}@.csc.es cheve spectculr reductons n communcton nd computton f compred wth the orgnl BnB-ADOPT + on severl enchmrks. Ths pper s orgnzed s follows. In secton we telegrphclly descre the concepts used n the rest of the pper (we ssume some fmlrty wth BnB-ADOPT nd soft rc consstency versons). We present our pproch n secton, dscussng some dfferences wth the centrlzed cse. We ntroduce the new lgorthms wth some detl n secton 4, nd ther expermentl evluton n secton 5. Fnlly, we conclude n secton 6. PRELIMINARIES COP. A nry Constrnt Optmzton Prolem (COP) s defned y (X, D, C), where X = {x,..., x n} s set of vrles; D = {D,..., D n} s collecton of fnte domns; D s the ntl domn of x ; C s set of unry nd nry soft constrnts represented s cost functons; C C specfes the cost of every comnton of vlues of vr(c ) = (x, x ), C : D D N {0, }. The cost of complete tuple s the ddton of ll ndvdul cost functons evluted on tht prtculr tuple. Ths defnton ssumes the weghted model of soft constrnts [5]. An optml soluton s complete tuple wth mnmum cost. Soft Arc Consstency. Let e nry COP: (, ) mens x tkng vlue, s the lowest uncceptle cost, C s the nry cost functon etween x nd x, C s the unry cost functon on x vlues, C φ s zero-ry cost functon tht represents necessry glol cost of ny complete ssgnment. As [], we consder the followng locl consstences (vrles re totlly ordered): Node Consstency*: (, ) s node consstent* (NC ) f C φ + C () < ; x s NC f ll ts vlues re NC nd there s D s.t. C () = 0; COP s NC f every vrle s NC. Arc consstency*: (, ) s rc consstency (AC) wrt. cost functon C f there s D s.t. C (, ) = 0; s support of ; x s AC f ll ts vlues re AC wrt. every nry cost functon nvolvng x ; COP s AC f every vrle s AC nd NC. Drectonl rc consstency*: (, ) s drectonl rc consstent (DAC) wrt. cost functon C, >, f there s D s.t. C (, ) + C () = 0; s full support of ; x s DAC f ll ts vlues re DAC wrt. every C, > ; COP s DAC f every vrle s DAC nd NC. Full D: COP s FDAC f t s DAC nd AC. AC /DAC cn e reched forcng supports/full supports to NC vlues nd prunng vlues not NC. Supports cn e forced on every vlue y proectng the mnmum cost from ts nry cost functons to ts unry costs, nd then proectng the mnmum unry cost nto C φ. Full supports cn e forced n the sme wy, ut frst t s needed to extend from the unry costs of neghors to the nry cost functons the mnmum cost requred to perform n the next step the pro-

2 ecton over the vlue. The systemtc pplcton of these opertons does not chnge the optmum cost nd mntns n optml soluton []. When we prune vlue from x to ensure AC /DAC, we need to recheck AC /DAC on every vrle tht x s constrned wth, snce the deleted vlue could e the support/full support of vlue of neghor vrle. So, deleted vlue n one vrle mght cuse further deletons n other vrles. The AC /DAC check must e performed untl no further vlues re deleted. DCOP. A Dstruted Constrnt Optmzton Prolem (DCOP) s defned y (X, D, C, A,α), where X, D, C defne COP, A = {,..., p} s set of p gents nd α : X A mps ech vrle to one gent. We ssume tht ech gent holds exctly one vrle (so vrles nd gents cn e used nterchngely) nd cost functons re unry nd nry only. Agents communcte through messges, whch could e delyed ut never lost, nd they re delvered n the order they were sent, for ny pr of gents. BnB-ADOPT. BnB-ADOPT [8] s reference lgorthm for DCOP. It s depth-frst verson of ADOPT [6], showng etter performnce. As ADOPT, t rrnges gents n DFS tree. Ech gent holds context, whch s set of ssgnments nvolvng the gent s ncestors, nd wll e updted wth messge exchnge. Messges re VALUE(,, vl, th), nforms chld or pseudochld tht t hs tken vlue vl wth threshold th, COST(k,, context, l, u) k nforms prent tht wth context ts ound re l nd u, nd TERMINATE(, ), nforms chld tht termntes. A BnB- ADOPT gent executes the followng loop: t reds nd processes ll ncomng messges nd tkes vlue. Then, t sends VALUE to ech chld or pseudochld nd COST to ts prent. BnB-ADOPT +. BnB-ADOPT + [] s verson of BnB-ADOPT tht sves most of redundnt VALUE nd COST messges, keepng optmlty nd termnton. BnB-ADOPT + cuses sustntl reductons n communcton costs wth respect to BnB-ADOPT. BnB-ADOPT + + SOFT ARC CONSISTENCY Here we present our contruton comnng dstruted serch (BnB- ADOPT + ) nd mntnng some knd of soft rc consstency for DCOP solvng. Due to the dstruted settng ths comnton requres some cre. In nve pproch, ech tme n gent needs nformton of other gent ths would generte two messges (request nd response) whch could cuse serous degrdton n performnce. In our pproch, we try to keep the numer of exchnged messges s low s possle, ntroducng the requred elements to enforce the selected soft rc consstency n exstng BnB-ADOPT + messges, keepng ther menngs for dstruted serch. Let us consder DCOP nstnce, where gents re rrnged n DFS tree nd ech gent executes BnB-ADOPT +. Let us consder generc gent self tht tkes vlue v. After sendng VALUE messges, self receves COST messges from ts chldren. A COST messge contns the lower ound computed y BnB-ADOPT +, wth the context (vrle, vlue) prs on whch ths lower ound ws computed. We consder COST messges whose context s smply the self gent wth ts ctul vlue v. If the sum of the lower ounds of these COST messges exceeds (the lowest uncceptle cost), the vlue v of self cn e deleted. To see ths, t s enough to relze tht the lower ound s computed ssumng (vrle, vlue) prs of context: f ths s smply (self, v), the ctul cost of v does not depend on the vlue of ny other gent, so f t exceeds t cn e deleted. Ths resonng s vld for ny gent. In ddton, some extr prunng cn e done t the gent locted t the root of the DFS tree (D root = {,,...}). Let us ssume tht ntlly root tkes vlue. After whle, root knows cost() = l() = u() =, nd t decdes to chnge ts ssgnment to. After exchngng some messges, root knows cost() = l() = u() =. If > then vlue cn e removed from D root ecuse cost() > cost(). Just removng wll cuse no effect n BnB-ADOPT +, ecuse t wll not consder gn s possle vlue for root. However, f we nform constrned gents tht s no longer n D root, ths my cuse some vlues of other gents to ecome unfesle so they cn e deleted. In these two cses, deletons re uncondtonl ecuse they do not depend on vlues of other gents. These deletons cn e further propgted n the sme wy, decresng the sze of the serch spce. Any deleton cused y propgton of uncondtonl deletons s lso uncondtonl. To propgte these deletons to other gents we need to mntn some knd of soft rc consstency durng serch. Mntnng soft rc consstency n the dstruted cse hs some dfferences wth the centrlzed cse. They re summrzed next: Prunng condton. In the centrlzed cse, vlue D cn e removed f t s not NC, tht s, f C () + C φ. However, n the dstruted cse cn e removed only f C () + C φ >, s explned n the followng. In oth cses, s n upper ound ( ) of the optmum cost. In the dstruted cse, BnB-ADOPT + termntes levng t ech gent n ssgnment tht elongs to soluton wth the optmum cost (optml soluton). Pruned vlues wll not e n ther domns when BnB-ADOPT + termntes. If we prune vlue when ts cost equls, we mght remove vlue tht elongs to n optml soluton. For ths reson, we cn 4 C ϕ =0 C ϕ =0 gent 0 C ϕ = gent Τ= C ϕ = 0 C ϕ = gent gent Fgure. Left: Smple exmple wth two gents nd nd two vlues per vrle. Bnry costs re ndcted, unry costs re zero. The optmum cost s nd there re two optml solutons (, )(, ) nd (, )(, ). Center: proects C on ts unry costs. No lnk etween two vlues of dfferent gents mens zero nry cost. Rght: f =, prunng v wth cost(v) = cuses to lose vlue (, ) whch s prt of the two optml solutons. In fct, no vlue remns for. Fgure. (UP) Left: Smple exmple wth two gents nd nd two vlues per vrle. Center: proects C on ts unry costs. Rght: proects unry costs on C φ. (DOWN) Center: proects C on ts unry costs, wthout consderng prevous proecton of (ths s ncorrect). Rght: proects unry costs on C φ, cusng n ncorrect ncrement.

3 BnB-ADOPT + messges: VALUE(sender, destnton, vlue, threshold) COST(sender, destnton, context[], l, u) STOP(sender, destnton) BnB-ADOPT + -AC messges: VALUE(sender, destnton, vlue, threshold,, C φ ) COST(sender, destnton, context[], l, u, sutreecontr) STOP(sender, destnton, emptydomn) DEL(sender, destnton, vlue) BnB-ADOPT + -FD messges: those of BnB-ADOPT + -AC plus UCO(sender, destnton, vectorofextensons) k DEL(,,) {,, c} DEL(,k,) D D k DEL(,,) UCO(,,[.,.,.,.]) {,, c} DEL(,k,) Fgure. Messges of BnB-ADOPT +, BnB-ADOPT + -AC nd BnB-ADOPT + -FDAC. only prune when the vlue cost exceeds. An exmple ppers n Fgure (n the centrlzed cse, the only gent executng the solvng procedure stores the complete est soluton found s serch progresses; vlue of the optml soluton cn e pruned from ts domn, ecuse tht soluton ws stored somewhere; when the lgorthm termntes, tht soluton wll e reclled). Legl representton of cost functons. In the centrlzed cse, ll cost functons re known nd mnpulted y sngle gent, the one n chrge of COP solvng. Ths gent keeps sngle copy of ech cost functon, where every updte s ccumulted. In the dstruted cse, cost functon C etween gents nd s known y oth gents, whch ntlly shre the sme representton of C. Opertons to mntn soft rc consstency modfy ths representton. Snce ech gent opertes dfferently, fter whle gents could hve dfferent representton of C. Both gents must mntn legl representton of C durng the soft rc consstency opertons. Otherwse, the sme cost cn e counted twce when proectng unry costs on C φ, s shown n Fgure, cusng C () + C φ to ecome n nvld lower ound for. To mntn legl representton, hs to smulte the cton of on ts C representton, nd vce vers. In some cses, hs lso to send messge to. In the dstruted cse, t s usully ssumed tht ech gent knows out () ts vrle nd () the cost functons t hs wth other gents. Assumpton () mples tht t lso knows out the domn of vrles t s constrned wth (ssumng tht cost functons do not contn rrelevnt vlues). To enforce ny soft rc consstency, we explctely requre tht f gent s connected wth gent y C, hs to represent loclly D. For prvcy resons, we ssume tht the unry costs of the vlues of n gent re held y tself, who knows them nd updtes them ccordng the locl consstency enforced. An gent nether cn know nor updte unry costs of other gents. Some soft rc consstences requre tht gents hve to e ordered. We tke the order of gents n ech rnch of the DFS tree used y BnB-ADOPT +. Oserve tht, lthough t s not totl order, gents n seprte rnches do not shre cost functons, so for enforcng soft rc consstency t s enough wth the orderng tht gents hve n DFS rnches. 4 BnB-ADOPT + AND AC /FDAC Dstruted serch cn cuse uncondtonl vlue deletons. These vlue deletons cn e propgted mntnng soft rc consstency Fgure 4. Three gents,, k n the sme rnch of the DFS tree. (Left) Mntnng AC : Cost functons re AC n oth senses; deletng vlue n D cuses to send two DEL messges to nd k to restore AC. (Rght) Mntnng FDAC : Cost functons re FDAC (DAC n one sense nd AC n the other); deletng vlue n D cuses to send two DEL messges to nd k to restore AC, plus one UCO messge to the hgher gent to restore DAC. durng dstruted serch. Ths de cn e esly ncluded n BnB- ADOPT +. Snce there re severl soft rc consstences, ths pproch genertes new lgorthms dependng on the selected soft rc consstency to e mntned. Here we present the connecton of BnB-ADOPT + wth AC nd FDAC. It s not dffcult to prove thn, no mtter mntnng AC or FDAC, the new lgorthms keep the optmlty nd termnton propertes of BnB-ADOPT [8]. 4. BnB-ADOPT + -AC BnB-ADOPT + -AC performs dstruted serch nd mntns AC level of soft rc consstency. If nd re two neghor gents, <, AC s mntned from to nd from to, s shown n Fgure 4 (left). Communcton etween gents s done y messge pssng. The semntc of orgnl BnB-ADOPT + messges remns unchnged. New elements re ncluded n these messges, they pper n Fgure. BnB-ADOPT + - requres some mnor chnges wth respect to BnB-ADOPT + : A new messge type, DEL, s requred. When self deletes vlue n D self, t sends DEL messge to every gent constrned wth t. Ths s depcted n Fgure 4 (left). When self receves DEL messge, t regsters tht the messge vlue hs een deleted from the domn of sender, nd t enforces on the constrnt etween self nd sender. If, s result of ths enforcng, some vlue s deleted n D self, t s propgted. VALUE messges nclude nd C φ. The ntl s pssed s prmeter nd root propgtes t downwrds, nformng the gents of the lowest uncceptle cost. As serch progresses, root my dscover lower vlues for, whch re propgted n the sme wy. Contrutons to C φ re propgted upwrds n COST messges nd ggregted n root, uldng C φ, lower ound of the nstnce glol cost (no mtter whch vlues re ssgned). Then, root propgtes C φ downwrds n VALUE messges. COST messges nclude the sutree contruton of ech gent to the glol C φ. Ech gent dds ts own contruton wth the sutree contrutons of ll ts chldren, nd the result s ncluded n the next COST messge sent to ts prent. All these contrutons re fnlly dded n root, formng the glol C φ, whch s propgted downwrds n VALUE messges.

4 procedure AC -preprocess( ) ntlze; AC (); whle end quescence do msg getmsg(); swtch(msg.type) DEL: ProcessDelete(msg); procedure AC () for ech neghors(self ) do f < self then AC -one-wy(self, ); AC -one-wy(, self ); else AC -one-wy(, self ); AC -one-wy(self, ); ST OP : ProcessStop(msg); procedure AC -one-wy(, ); /* fter executon, AC from to holds */ FromBnryToUnry(, ), f = self then PruneDomnSelf(); FromUnrySelfToC φ (); procedure FromBnryToUnry(, ) for ech D do v rgmn D {C (, )}; α C (, v); for ech D do C (, ) C (, ) α; f = self then C () C () + α; procedure FromUnrySelfToC φ () v rgmn Dself {C self ()}; α C self (v); mycontruton mycontruton + α; for ech D self do C self () C self () α; procedure PruneDomnSelf() for ech D self do f C self () + C φ > then DeleteVlue(); procedure DeleteVlue() D self D self {}; f D self = then for ech neghors(self ) do sendmsg:(stop, self,, true); end true; else for ech neghors(self ) do sendmsg:(del, self,, ); AC -one-wy(, self ); FromUnrySelfToC φ (); f = myvlue then myvlue rgmn v Dself LB(v); procedure ProcessDelete(msg) D sender D sender {msg.vlue}; 4 AC -one-wy(self, sender); procedure ProcessStop(msg) f (msg.emptydomn = true) then for ech neghors(self ), sender do sendmsg(stop, self,, true); end true; Fgure 5. The preprocess code for enforcng AC. We ssume tht cost functons re ntlly AC. If not, they re mde AC y preprocess of Fgure 5. A quck descrpton follows: AC-preprocess. It receves the ntl nd performs AC. Then, t performs recevng loop of DEL or STOP messges tht ends when n empty domn hs een detected (end s true) or when there re no more messges (quescence s true). AC (). For ech nry cost functon n whch self s nvolved, t enforces AC wth the followng ssumpton: t proects frst on the lower gent nd then on the hgher gent. It s worth notng tht executng AC -one-wy(, self ) does not chnge unry costs of self vlues, ut modfes the representton of C,self n self n the sme wy gent does. AC -one-wy(, ). It enforces AC property from to. FromBnryToUnry(, ). It proects nry costs C on unry costs. It updtes unry costs when the frst rgument s self. FromUnrySelfToC φ (). It proects self unry costs on mycontruton, whch ccumultes self contruton to C φ. PruneDomnSelf(). Checks for deleton every vlue n D self. DAC -one-wy(); do nothng /* */ f < self then DAC -one-wy( ); else AC -one-wy(, self ); 4 f self > sender AC -one-wy(self, sender); procedure DAC -one-wy() P[] mn Dself {C,self (, ) + C self ()}; E[] mx D {P[] C,self (, )}; sendmsg(uco, self,, E); FromUnrySelfToBnry(, E); FromBnryToUnry(, self ); procedure FromUnrySelfToBnry(, vector) for ech D self do for ech D do C,self (, ) C,self (, ) + vector[]; C self () C self () vector[]; procedure ProcessUnryCosts(msg) for ech D sender do for ech D self do C self,sender (, ) C self,sender (, ) + msg.vector(); /* extenson */ FromBnryToUnry(self, sender); PruneDomnSelf(); FromUnrySelfToC φ (); for ech neghors(self ) do f < self then DAC -one-wy(self, ); Fgure 6. Replcng lnes,,, 4 of Fgure 5 for the ones ndcted here, we otn the preprocess code for enforcng FDAC. When UCO messge rrves, ProcessUnryCosts(msg) s clled. DeleteVlue(). self removes vlue from D self. If D self =, there s no cceptle soluton, so self sends STOP messges to ll ts neghors, ndctng tht the process termntes. Otherwse, for ll neghors, DEL messge s sent notfyng deleton nd AC -one-wy(, self ) s executed. Oserve tht ths cuses no chnge n self unry costs, whch re proected on C φ. If the deleted vlue ws the current vlue, new vlue s selected. ProcessDelete(msg). self receved DEL messge: sender hs deleted vlue from D sender. self regsters ths n ts D sender copy nd enforces AC from self to sender. ProcessStop(). self receved STOP messge. If cused y n empty domn, self resends the STOP messge to ll ts neghors, except sender. In ny cse, self records ts recepton n end. The BnB-ADOPT + -AC process code s not gven here for spce resons. It s sed on BnB-ADOPT + []. In ddton to the norml BnB-ADOPT + operton, t ncludes the followng ctons to mntn AC. When self receves VALUE messge, the locl copes of nd C φ re updted f the vlues contned n the receved messge re etter (lower for, hgher for C φ ). If or C φ chnged, D self s tested for possle deletons (ecuse elements of the deleton condton hve chnged). When self receves COST messge from chld c, self records c sutree contruton to C φ. In the Bcktrck procedure, when self chnges vlue, D self s tested for possle deletons. When self receves DEL messge, the procedure ProcessDelete(msg) tht ppers n Fgure 5 s clled. 4. BnB-ADOPT + -FDAC BnB-ADOPT + -FDAC performs dstruted serch nd mntns FDAC level of soft rc consstency. If nd re two neghor gents, <, DAC s mntned from to nd AC from to, s shown n Fgure 4 (rght). As ndcted n Fgure, n ddton to the messges requred for BnB-ADOPT + -AC, t requres the new UCO (unry costs) messge. When self enforces DAC on cost functon wth hgher gent, self sends UCO messge to wth

5 the mnmum contruton of self unry costs for to proect on unry costs (followng []). Ths s depcted n Fgure 4 (rght). It s worth notng tht ths DAC enforcng does not elmntes prevous AC enforcng on the sme pr of gents, theorem of []; we lwys enforce AC efore enforcng DAC on C. The vector of extensons s the E[] computed n the procedure DAC -one-wy() n Fgure 6. Upon recepton, the gent wll perform the extenson of these unry costs nto the nry cost functon, the proecton of the nry costs nto the unry ones nd these on C φ, checkng ts domn for possle deletons nd restorng the DAC condton from towrds hgher neghors. We ssume tht cost functons re ntlly FDAC. If not, they cn e mde FDAC y the preprocess depcted n Fgure 6, where lnes,,, 4 replce the correspondng ones n Fgure 5. A summry descrpton of ths code follows: Insted of AC, self enforces DAC wth hgher gent. self does nothng ecuse to enforce DAC wth lower gent, self hs to wt for the UCO messge. self enforces ether AC or DAC, dependng on the reltve order etween nd self. 4 self enforces AC wth the hgher gent sender. DAC -one-wy(). self strts enforcng DAC on C,self y performng the requred opertons on ts representton of C,self nd sendng UCO messge to. FromUnrySelfToBnry(, vector). self dds n C,self the costs n vector tht wll e sent to, sutrctng them from C self unry costs. ProcessUnryCosts(msg). self receves the UCO messge nd extends ts costs nto C self,sender. It proects costs from C self,sender on ts unry costs nd these on C φ. self tres to prune ts domn nd enforces DAC wth ny other hgher gent constrned wth t. The BnB-ADOPT + -FDAC process code s not gven here for spce resons. Bsclly t s the BnB-ADOPT + -AC code, plus the recepton nd process of the new UCO messge. Ths process s done y ProcessUnryCosts(msg) tht ppers n Fgure 6. 5 EXPERIMENTAL RESULTS We evlute the effcency of BnB-ADOPT + -/FD y dscrete event smultor. Performnce s evluted n terms of communcton cost (messges exchnged), computton effort (nonconcurrent constrnt checks), consderng lso the numer of tertons (synchronous cycles; n cycle every gent reds ll ts ncomng messges, processes them nd sends ll ts outgong messges) the smultor must perform untl the soluton s found. We tested our lgorthms on unstructured nstnces wth nry rndom DCOPs, nd on structured dstruted meetng schedulng dtsets. Bnry rndom DCOP re chrcterzed y n, d, p, where n s the numer of vrles, d s the domn sze nd p s the network connectvty. We hve generted rndom DCOP nstnces: n = 0, d = 0, p = 0., 0.4, 0.5, 0.6. Costs re selected from n unform cost dstruton. Two types of nry cost functons re used, smll nd lrge. Smll cost functons extrct costs from the set {0,..., 0} whle lrge ones extrct costs from the set {0,..., 000}. The proporton of lrge cost functons s /4 of the totl numer of cost functons (ths s done to ntroduce some vrlty mong tuple costs; usng unque type of cost functon cuses tht ll tuples look pretty smlr from n optmzton vew). Results pper n Tle (), verged over 50 nstnces. On the meetng schedulng formulton, vrles represent meetngs, domn represent tme slot ssgned for ech meetng, nd there re constrnts etween meetngs tht shre prtcpnts. We present 4 cses otned from the DCOP repostory [9] wth dfferent herrchcl scenros nd domn 0: cse A (8 vrles), cse B (0 vrles), cse C ( vrles) nd cse D ( vrles). Results pper n Tle (), verged over 0 nstnces. For ech prolem, we clculte n ntl to hve prune opportuntes on the nd FD preprocess. Ths s done n the followng wy. Ech lef gent choose the est vlue wth locl nformton, nd nforms ts prent of the selected vlue nd ts cost. Prents receve ths nformton from chldren nd choose ther own est vlue regrdng locl nformton, nd lso nform ther prents ccumultng the cost of the prtl soluton. When ll gents hve chosen ther vlue, we hve complete soluton (lkely not the optml one) whch s n upper ound of the optmum prolem cost. So root clcultes the cost of ths complete soluton nd nform ths cost downwrds. Ths cost s consdered the ntl of the prolem. Wth ths preprocess we re le to clculte dfferent from, requrng only two messges per ech gent: one from chld to prent nformng the prtl soluton cost, nd one from prent to chldren nformng of the glol ntl. On rndom DCOPs, BnB-ADOPT + -/FD showed cler enefts on communcton costs wth respect to BnB-ADOPT +. Mntnng level (BnB-ADOPT + -) the numer of exchnged messges s dvded y fctor from to 0. Notce tht ths reducton s otned genertng only very few DEL messges. In ddton, ncludng the FD level (BnB-ADOPT + -FD) enhnces ths reducton, dvdng the numer of BnB-ADOPT + exchnged messges y fctor from 5 to 7. Notce tht mntnng the hgher FD level ncreses slghtly the numer of DEL messges (ths s ecuse more deletons hve een generted) nd only very few UCO messges re dded. In contrst, mportnt svngs re otned compred to. In generl, ncludng few DEL nd UCO messges nd performng extr locl computton to enforce soft rc consstency llows BnB-ADOPT + -/FD to otn lrge reductons n VALUE nd COST messges. Ths s ecuse vlues tht wll not e n ny optml soluton whch would e dscovered y dstruted serch re sooner removed y soft rc consstency, so gents wll need to ssgn less vlues (consequently they wll generte less VALUE messges) when testng the optmum ssgnment for ech context. If less VALUEs re generted, less COSTs wll e sent n response. We ssume the usul cse where communcton tme s hgher thn computton tme, then the totl elpsed tme s domnted y the communcton tme, nd reducng the numer of messges cuses n mportnt effect n performnce. We lso oserve cler decrement n the numer of cycles of BnB- ADOPT + -/FD (dvded y fctor from to 7), comned wth decrement n the numer of messges per cycle wth respect to the orgnl BnB-ADOPT +. Assumng tht processng ech messge type requres pproxmtely the sme tme, the comnton of these two effects s n mprovement ndctor. Snce gents need to process less nformton comng from ther neghors on ech terton, nd they perform less tertons to rech the optmum, ths comned reducton s very enefcl for gent performnce. Notce tht lthough gents need to perform more locl computton to mntn locl consstency, the numer of non-concurrent constrnt checks (NCCCs) lso shows mportnt reductons. Ths s the comnton of two opposte trends: gents re dong more work enforcng soft rc consstency nd processng new DEL nd UCO messges, ut less work processng less VALUE nd COST mes-

6 () Rndom DCOPs p #Msgs #VALUE #COST #DEL #UCO #Cycles #NCCC #Deletons 54,45 9,007 5, , 7,7,7 0 5,0 8,489 6, ,96 705, ,50 5,564,54 45,95 575, ,74,888,40,8 4,, ,566,07,9 0 6,4 96,087 5, ,47 0,979, ,4 05,50 0, ,59 5,69, ,680,458,69,8 6,986, ,09 5,86,055 0,469,4,09,84,76, ,7,84, ,87,05 57, 75, ,496 66,967,0 7 7,8,885,,80 5,480, ,06 89,6,84 0,99,84,0,889,75, ,408,987, ,7,74 64,6 69, ,8 5,705,844 7 () Dstruted Meetng Schedulng #Msgs #VALUE #COST #DEL #UCO #Cycles #NCCC #Deletons 5,767 4,0, ,47 690, ,88,46, ,06 0,040 4 A 5,5,98, ,67 0, ,45 8,8 40, ,50 80,84 0,474 4,94 6,69 5 0,585,54 44 B 0,07 4,7 5, ,6 97,964 5,86 6,907 6, ,78 57,995 0,55,655, , C,990,49, , ,86 9,457 0,97 0 0,7 4,86 0,507,708, ,4 74 D,96,474, , Tle. Expermentl results of BnB-ADOPT + (frst row) compred to BnB-ADOPT + - (second row) nd BnB-ADOPT + -FD (thrd row) sges. Ths comnton turns out to e very enefcl, svng computtonl effort for ll cses tested. In some cses, reducton reches up to one order of mgntude. For the meetng schedulng nstnces, we lso otn cler enefts mntnng, enhnced y FD. For the stronger FD level (BnB-ADOPT + -FD) messges re dvded y fctor from 4 to 6, cycles re dvded y fctor from to 4 nd there re sgnfcnt svngs n NCCCs. To otn these results, very few DEL nd UCO messges re needed, nd the extr computtonl effort requred to mntn or FD s effectvely lnced y the decrement on VALUE nd COST messges. So, mntnng soft rc consstency (BnB-ADOPT + - AC /FDAC ) proved to e clerly enefcl for the nstnces tested. The propgton of deletons contrutes to dmnsh the serch effort, decresng the numer of COST nd VALUE messges exchnged. Also, the flows of costs from one gent to nother, mplemented y UCO messges, llows n gent to pss some of ther untry costs to hgher gents, serchng for more prunng opportuntes. In the worst cse, mntnng FDAC our pproch dvdes the numer of messges requred to rech n optml soluton y fctor of, sustntlly decresng the numer of cycles nd the computtonl effort t ech gent. 6 CONCLUSION In ths work we hve connected BnB-ADOPT + wth some forms of soft rc consstency n the weghted cse, mng t detectng nd prunng vlues whch would not e n the optml soluton, wth the fnl gol of mprovng serch effcency. These deletons re uncondtonl nd do not rely on ny prevous vrle ssgnment. The trnsformtons ntroduced (extendng unry costs nto nry ones, proectng nry costs nto unry ones, proectng unry costs nto C φ, nd prunng vlues not NC ) ssure tht the optmum (nd ny optml soluton) of the trnsformed prolem remns the sme s the orgnl nstnce. Accordng to expermentl results, propgton of uncondtonl deletons provdes sustntl enefts for the enchmrks tested. New messges DEL nd UCO hve een ntroduced. However, the ncrement n the numer of messges due to the generton of new DEL nd UCO messges hs een lrgely compensted y the decrement n the numer of COST nd VALUE messges used to solve the prolem. BnB-ADOPT + -AC /FDAC hs een proved to e very enefcl wth respect to BnB-ADOPT +, not only n communcton cost ut lso n computton effort. ACKNOWLEDGEMENTS Ths work s prtlly supported y the proect TIN C0-0. We wnt to thnk the referees for ther constructve comments. REFERENCES [] R. Ber, C. Fernndez, M. Vlls, C. Domshlk, C. Gomes, B. Selmn, nd B. Krshnmchr, Sensor networks nd dstruted csp: Communcton, computton nd complexty, Artfcl Intellgence, 6, 7 47, (005). [] P. Guterrez nd P. Meseguer, Svng messges n BnB-ADOPT, Proc. AAAI-0, (00). [] J. Lrros nd T. Schex, In the quest of the est form of locl consstency for weghted CSP, Proc. of IJCAI-0, (00). [4] J. Lrros nd T. Schex, Solvng weghted csp y mntnng rc consstency, Artfcl Intellgence, 59, 6, (004). [5] P. Meseguer, F. Ross, nd T. Schex, Hndook of Constrnt Progrmmng. Chpter 9, Soft Constrnts., Elsever, 006. [6] P. J. Mod, W.M. Shen, M. Tme, nd M. Yokoo, Adopt: synchronous dstruted constrnt optmzton wth qulty gurntees, Artfcl Intellgence, 6, 49 80, (005). [7] R. Wllce nd E. Freuder, Constrnt-sed resonng nd prvcy/effcency trdeoffs n mult-gent prolem solvng, Artfcl Intellgence, 6, 09 7, (005). [8] W. Yeoh, A. Felner, nd S. Koeng, Bn-dopt: An synchronous rnch-nd-ound DCOP lgorthm, Proc. of AAMAS-08, , (008). [9] Z. Yn. USC dcop repostory. Meetng schedulng nd sensor net dtsets,