A Partial Decision Scheme for Local Search Algorithms for Distributed Constraint Optimization Problems

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1 A Partal Decson Schee for Local Search Algorths for Dstrbuted Constrant Optzaton Probles Zhepeng Yu, Zyu Chen*, Jngyuan He, Yancheng Deng College of Coputer Scence, Chongqng Unversty, Chongqng, Chna { , chenzyu, ABSTRACT Local search algorths are wdely adopted n solvng large-scale Dstrbuted Constrant Optzaton Probles (DCOPs). However, snce each agent always aes ts value decson based on the values of ts neghbors n local search, those algorths usually suffer fro local preature convergence. More concretely, an agent cannot ae a wse decson wth poor values of ts neghbors snce ts decson space s bound up wth those poor values. n ths paper, we propose a Partal Decson Schee (PDS) to relax the decson space of an agent by gnorng the value of ts neghbor whch has the bad pact on ts local benefts. The PDS coprses two partal decson processes: trgger partal decson and recursve partal decson. The forer s teratvely perfored by agents who cannot enhance ther local benefts unlaterally to brea out of potental local opta. The latter s recursvely perfored by neglected agents to prove global benefts. Besdes, the trgger condtons along wth a self-adaptve probablty are ntroduced to control the use of PDS. The PDS can be easly appled to any local search algorth to overcoe ts local preature convergence wth a sall addtonal overhead. n our theoretcal analyss, we prove the feasblty and convergence of PDS. Moreover, the experental results also deonstrate the superorty of the use of PDS n the typcal local search algorths over state-of-the-art local search algorths. Author Keywords: Mult-agent Syste; Dstrbuted Constrant Optzaton Proble; ncoplete Algorth; Local Search Algorth; Partal Decson. NTRODUCTON Wth the advent of dstrbuted artfcal ntellgence, ult-agent systes (MAS) [] have becoe a popular way to odel the coplex nteractons and coordnaton requred to solve dstrbuted probles. Dstrbuted Constrant Optzaton Probles (DCOPs) are a fundaental fraewor for odelng ult-agent coordnaton probles, wdely deployed n real applcatons such as tas schedulng [2,3], resource allocaton [4], sensor networs [5,6], etc. A DCOP conssts of a set of agents, each holdng one or ore varables. Each varable has a doan * correspondng author. Appears n: Proc. of the 6th nternatonal Conference on Autonoous Agents and Multagent Systes (AAMAS 207), S. Das, E. Durfee, K. Larson, M. Wnoff (eds.) May 8 2, 207, São Paulo, Brazl. Copyrght 207, nternatonal Foundaton for Autonoous Agents and Multagent Systes ( All rghts reserved. of possble value assgnents. Constrants aong varables assgn costs to cobnatons of value assgnents [7]. The agents ust coordnate ther decsons of value assgnents so that the su of all the constrants s optzed. Algorths to solve DCOPs can be classfed as beng ether coplete [8-2] or ncoplete, based on whether they can fnd the optal soluton or they get suboptal solutons wth sall executon te. However, snce DCOPs are NP-hard [7,8], coplete algorths ncur exponental councaton or coputaton wth the ncrease of scale and coplexty of probles, whch lts ther use n any real applcatons. ncoplete algorths whch requre very lttle local coputaton and councaton to fnd suboptal solutons can be well appled to large-scale practcal dstrbuted applcatons. Therefore, there has been growng nterest n the last few years n ncoplete DCOP algorths. ncoplete algorths generally nclude the followng three categores [3]: local search algorths, nference-based algorths and saplng-based algorths. nference-based ncoplete algorths le Max-Su [4] and ts varants [5,6] allow agents to explot the structure of a constrant graph to aggregate rewards fro ther neghbors but are ore approprate to acyclc DCOP graphs. Saplng-based ncoplete algorths le DUCT [7] and D-Gbbs [8] saple the search space to approxate a functon as a product of statstcal nference [3]. Local search algorths are the ost popular ncoplete ethods for DCOPs, where each agent optzes based on ts local constrants and the values receved fro all ts neghbors, such as DSA [9], DBA [9,20] and MGM [2]. However, Local search algorths are prone to converge to local opta. DSAN [22] tres to prove DSA by sacrfcng ndvdual benefts restrcted by the neghbor values. But the algorth s ore sutable for Dstrbuted Constrant Satsfacton Probles (DCSPs). K-optalty [23,24] was proposed to prove the soluton of local convergence by coordnatng the decson of all agents wthn the K-sze coalton, such as MGM-2 [2], MGM-3 [25] and KOPT [26]. However, one of the dffcultes n K-optalty algorths nvolves the defnton of K. Moreover, wth the ncrease of K, these algorths requre greater coputatonal effort to fnd a K-optal soluton [25]. Recently, an anyte local search (ALS) fraewor wth soe exploraton heurstcs [7] are presented to enhance local search algorths. For exaple, DSA-PPRA uses a perodc ncrease n the level of exploraton and DSA-SDP eploys an exploratve probablty wth regard to the potental proveent. Besdes, rando restart was ntroduced nto the ALS fraewor as an algorth-ndependent heurstc to facltate exploraton n ncoplete DCOP algorths. By analyzng the local search process, we fnd that the decson space of an agent s parttoned n ters of the values of ts neghbors snce ts value decson heavly reles on the values of ts neghbors. f always recevng the sae values fro ts 87

2 neghbors, an agent wll get nto the sae decson parttons and have no opportunty to search other prosng parttons. That wll lead to local preature convergence. The paper presents a partal decson schee (PDS) to enable an agent to search the prosng decson parttons by gnorng soe bad value of ts neghbor to enhance ts local benefts, and coordnate the decson of ts neglected neghbor to prove global benefts. The PDS breas the assgnent dependence on the neghbor values and help local search to escape fro local opta. The paper s organzed as follows. n Secton 2, we present the DCOP defnton and the local search fraewor. Secton 3 llustrates the detals of our proposed schee (PDS). Secton 4 proves the feasblty and convergence of PDS. n Secton 5, an experental study s presented to evaluate PDS when cobned wth the typcal local search algorths n coparson wth exstng local search algorths. Fnally, Secton 6 concludes ths paper. 2. Bacground 2. Dstrbuted Constrant Optzaton Probles A DCOP can be represented by a tuple A, X, D, C s a set of agents; X x x x such that: A a, a2,, a, 2,, n s a set of varables, where each varable s assgned to an agent; D D, D2,, Dn s a set of fnte and dscrete doans, where D s the doan of varable x; C c c c,,..., 2 constrants, where each constrant c : D D q s a set of + specfes a non-negatve cost for every possble value cobnaton of a set of varables [7,8]. Gven ths, the goal for the agents s to fnd the ont varable assgnent X* such that a gven global obectve functon s nzed. Generally, the obectve functon s descrbed as the su over C. To facltate understandng, ths paper assues that each agent has a sngle varable and constrants are bnary relatons. Here, the ter agent and varable can be used nterchangeably. A bnary constrant s a constrant nvolvng exactly two varables defned as c : D D +. The ont varable assgnent X* s obtaned as follows: x3 X* arg n c ( x v, x v ) () x x2 x4 (a) constrant graph v D, v D c C (b) constrant atrces Fgure. A DCOP nstance t s coon that a DCOP proble s vsualzed as a constrant graph where the nodes are the agents and the edges are the constrants. Fgure shows an exaple of a DCOP proble whose constrant graph and constrant atrces are shown n Fgure (a) and Fgure (b), respectvely. The DCOP nstance ncludes 4 varables, each of whch taes a value n {, 2}, and the cells of the constrant atrces contan the costs of the assgnent n Fgure (b). 2.2 Local Search Fraewor for DCOPs The general desgn of local search algorths for DCOPs s synchronous. n each round of a basc local search fraewor, an agent sends ts value to all ts neghbors n the constrant graph and receves the values of ts neghbors. Then, the agent wll select a value n ters of the values of ts neghbors and decde whether to replace ts value accordng to a replaceent strategy. The an dfference aong local search algorths s replaceent strategy. For exaple, agents n DSA stochastcally replaces ther value every round f the replaceents can reduce ther local costs whle only agents wth axal gans aong ther neghbors can replace ther values n MGM. Here, we only present one algorth that apples to ths general fraewor, the Dstrbuted Stochastc Algorth (DSA). Algorth : Dstrbuted Stochastc Algorth (DSA) For each agent x executes:. value Choose_Rando_Value( ) 2. send value to neghbors 3. whle (no ternaton condton s et) 4. collect neghbors value 5. select a new value whch reduces the local cost ost 6. Δ the nuber of the cost reduced by the new value 7. f (Δ > 0 and rando( ) < p) 8. assgn the new value 9. send value to neghbors Fgure 2. A fraewor of DSA The basc dea of the DSA algorth s sple. A setch of DSA s presented n Fgure 2. An agent x starts wth an ntal process n whch x assgns a rando value and sends the value to all ts neghbors (lne -2). Then, x perfors a sequence of rounds untl the ternaton condton s et. n each round (lne 4-9), x collects the assgnents of ts neghbors and selects a new value to reduce the local cost ost. Then, t decdes, often stochastcally, to eep the current value or change to the new one, f Δ > 0 (see [9] for detals on the possble strateges). n ths paper, we call the process n each round local search. 3. Partal Decson Schee 3. Motvaton Let us tae Fgure as an exaple to llustrate the atter of local convergence and the dea of the proposed schee when DSA s carred out. Assue that all agents assgn as after any rounds and send ther value to ther neghbors n the next round. x wll receve x2= and x3=, and try to select a new value to reduce the current local cost (lne 4-6) n ters of ts decson parttons pertanng to x2= and x3= (.e., the whte coluns of the constrant atrces of x shown n Fgure 3). n other words, x has no chance to search other parttons (.e., the gray coluns shown n Fgure 3). x wll not change ts value (lne 7 wll be false) snce ts value results n the nal local cost 5 based on the decson partton. However, t can be observed fro the constrant atrces of x that the best value of x s 2 wth the nal local cost 3 f x2 and 88

3 x3 select 2 and, respectvely. Unfortunately, x wll not change ts value unless x2 and x3 change ther values. Slarly, x2, x3 and x4 wll not change ther values upon recevng fro ther neghbors accordng to the constrant atrces n Fgure (b). At ths pont, DSA converges to a local nu. x x 2 2 x x Fgure 3. The decson partton of x wth x2= and x3= x x 2 2 x x Fgure 4. The decson partton of x after gnorng x2= We hope to escape fro a local nu by breang the assgnent dependences aong the neghbors. Our dea s to enable an agent to search the prosng decson parttons by gnorng soe bad value of ts neghbor so as to enhance ts local benefts. For exaple, x can search ore decson parttons and get the n cost 3 when gnorng x2=, shown n Fgure 4. However, to acheve ts proveent, an agent who neglects ts neghbor ust provde the suggested value and ts gan to the neglected neghbor. The neglected neghbor wll decde to accept or refuse the suggeston based on ts local benefts and the receved gan to ensure the proveent of global benefts. n ths exaple, x can get the local cost 3 by gnorng x2= or 9 by gnorng x3=. Therefore, x wll gnore x2= and change ts own value to 2 under the assupton that x2=2. Meanwhle, t wll nfor the suggeston (x2=2 and gan=2) along wth ts new value to x2. Based on ts constrant atrces and the receved gan, x2 wll get the accuulated gan by acceptng x2=2 and gnorng x4=. So, x2 wll change ts value to 2 and nfor the suggeston (x4=2 and gan=2) along wth ts new value to x4. Eventually, x, x2, x3 and x4 tae 2, 2, and 2, respectvely, whch s the best soluton to the exaple. On the bass of the above procedure, we propose a partal decson schee (PDS). 3.2 Fraewor of Partal Decson Schee The PDS ncludes two partal decson processes, trgger partal decson and recursve partal decson. t can be easly appled to any local search algorth (the pseudocode s shown n Fgure 5). Trgger condtons. As shown n the exaple of Subsecton 3., partal decson should be perfored when an agent converges to a local nu. n other words, only f an agent cannot prove ts state by perforng the orgnal local search schee, partal decson should be trggered; otherwse, the orgnal local search ght be broen and cannot coe nto play when ntroducng partal decson. However, t s hard for an agent to dentfy whether the current local search process has gotten stuc n a true local nu due to the lac of global nforaton. Therefore, referrng to quas-local nu (QLM) n [27], we ntroduce potental-local nu (PLM) and a self-adaptve partal decson probablty (PDP) to control the use of PDS. Algorth 2: PDS for a local search algorth For each agent x executes: 0. ntalze Local Search Algorth. whle (no ternaton condton s et) 2. f recevng suggeston_essage{v*s,gs} 3. Dspose_suggeston_essage(v*s,gs) 4. calculate probablty PDP 5. create a rando nuber r (0< r <) 6. f x s n a PLM and r < PDP 7. Trgger_partal_decson( ) 8. else 9. perfor Local Search Trgger_partal_decson( ) 20. select a neghbor x randoly 2. calculate value v* and v* by forula (8) and (9) 22. calculate the n local cost c* by forula (0) 23. g c c* 24. f (g > 0) 25. assgn value v* 26. send suggeston_essage{ v*,g } to x Dspose_suggeston_essage(v*s,gs) 27. v* v*s 28. calculate the unon gan gu by forula (2) 29. f (gu > 0) 30. assgn value v* 3. go to lne 32. else 33. Recursve_partal_decson( ) Recursve_partal_decson( ) 34. fnd x by forula (3) 35. calculate value v* by forula (4) 36 calculate g by forula (5) 37. f (g > 0) 38. assgn value v* 39. send suggeston_essage{v*,g} to x 40. go to lne Fgure 5. A PDS-based local search Algorth executed by x Defnton : An agent x s n a potental-local nu (PLM) f x cannot reduce ts cost unlaterally by the orgnal local search schee n ters of the receved values fro all ts neghbors. Here, PLM descrbes a state whch s lely to be or evolve nto a true local nu. t plays the sae role as QLM. However, PLM s a weaer condton than QLM snce an agent detects a PLM only usng ts own current nforaton rather than nforaton about ts neghbors. Thus, PLM s easer to be detected by an agent, whch eans the tendency of local na can be perceved earler. We also tested the effect of PLM and QLM on our schee and found that the algorths wth PLM perfored slghtly better than the ones wth QLM n ters of soluton qualty n the experent. When an agent detects that t s n a PLM, t trggers a partal decson wth a partal decson probablty (PDP) (lne 6-7). Here, the PDP s forulated n ters of the qualty of the current local soluton and the rounds. n order to evaluate the current soluton for x, a local cost level L s defned as follows: c cn, c L c ax cn 0, c ax ax c c n n (2) 89

4 c c ( x v, x v ), v D, v D (3) N c n c ', arg n n ', ' ' x v x c v D v' x v x v (4) D N c ax ax c ', arg ax ', ' ' x v x c v D v' x v x v (5) D N Here, v and v are the current value of x and ts neghbor x, respectvely. c s the local cost of the current soluton for x. cn and cax are the lower and upper bounds of the local cost of x, respectvely. N s an ndex set of all neghbors of x. t can be seen fro forula (2) that the greater the L, the worse the current local soluton. Addtonally, x wll record the nal L coputed so far. To accoodate the search process, a PDP for x s defned as the followng two fors: M _ round C _ round PDP L (6) M _ round M _ round C _ round PDP L (7) M _ round Here, C_round and M_round represent the current and ax rounds, respectvely. Max rounds are the axal synchronc teratons whch are usually used as a ternaton condton of the local search. ntally, PDP s calculated accordng to forula (6). Once the optal L retans unchanged for T rounds, PDP s calculated accordng to forula (7). n the paper, T= M_round /0. t can be nferred fro forula (6) and (7) that x wth greater L has a hgher probablty to trgger a partal decson and the nfluence of PDS wll gradually wane wth the ncrease of the rounds. Trgger Partal Decson ntates partal decson and wll be perfored once the trgger condtons are et. When x eets PLM and PDP, t wll call Trgger_partal_decson( ) (Lne 20-26). Frstly, x randoly selects one of ts neghbors, x (Lne 20). Then, x calculates the new value v*, v* and the nal local cost c* by forula (8), (9) and (0), respectvely (lne 2-22). v* and v* are the values of x and x correspondng to c* wthout regard to the receved value fro x, respectvely. v s the receved value fro a neghbor x. * arg n ( ', ) n ( ', ' ) v c ' x v x v v c ' x v x v (8) D v D N v* argn c ( x v*, x v' ) (9) v' D c* c ( x v*, x v ) c ( x v*, x v* ) (0) N g=c-c* () Then, x calculates ts gan g (lne 23) by forula () and decdes whether to change ts current value to v* by the followng process naed decson and replaceent (lne 24-26): f g s no greater than 0, whch eans x cannot prove ts state when gnorng x, x wll retan ts current value and contnue to do the next round of local search; otherwse, x wll change to v* and send a suggeston essage to ts neghbor x. The suggeston essage ncludes the suggested value v* and the gan g. Dspose Suggeston Message s perfored by the neglected agent to decde whether to accept the suggested value or to further perfor partal decson. Upon recevng a suggeston essage, x wll execute Dspose_suggeston_essage(v*s, gs). Here, gs and v*s s the gan and the suggested value n the suggeston essage sent by xs, respectvely. x frstly records v*s as v* and then calculates ts unon gan gu by forula (2) (lne 27-28): g g ( c ( x v, x v ) c ( x v*, x v )) (2) u s N, s f gu s no less than 0, x wll accept the suggeston by changng ts current value to v* and sp the current round of local search to avod the conflct between local search and partal decson (lne 29-3); otherwse, x wll perfor recursve partal decson (lne 32-33). Recursve Partal Decson s perfored by the neglected agent to prove global benefts by gnorng one of ts neghbors except ts suggester. Frstly, x searches ts decson space by gnorng the value of each neghbor except xs to fnd a neghbor x whch could nze x s cost f t changed ts value v accordng to forula (3) (lne 34). Then, x coputes the new value v* for x by forula (4) and the gan g of all ts constrants except cs by forula (5) (lne 35-36). Here, cs has been consdered n gs n the suggeston essage. argn c ( x v *, x v ) n c ( *, ' ) v' x v x v N D N s, s (3) v* argn c ( x v*, x v' ) (4) v' D g c ( x v, x v ) N s (5) c( x v *, x v ) c ( x v *, x v * ) N s, Next, x perfors the decson and replaceent process shown by Trgger Partal Decson. Ths ay recursvely trgger new recursve partal decson for an agent x recevng the suggeston essage untl x eets gu > 0 or g 0 (lne 37-40). t s worth notng that we use the dfferent strateges to select a neglected neghbor, x. n trgger partal decson, x s randoly selected snce the rando selecton requres only lttle coputaton and can avod selectng the sae neghbor repeatedly so as to enhance the dversty whch s an portant concern n trgger partal decson. n recursve partal decson, x s optally selected snce the best choce can guarantee the proveent of global beneft. Addtonally, the PDS requres a sall overhead. Snce the calculaton for L requres O() te, and cn and cax can be coputed n preprocessng, the overhead anly concentrate n two parts: the extra essage nuber and the te and space coplexty caused by trgger partal decson and recursve partal decson. Actually, the nuber of extra essages equals to the 90

5 nuber of agents perforng partal decson. And, each agent x requres O( D ( N ax D )) space n te and N O D N trgger partal decson, and O( N * ax N D ) te and O( N ) space n recursve partal decson, whch s close to the coplexty of local search. Moreover, wth the trgger condtons, there s only a sall set of agents executng partal decson. Thus, the overall overhead s sall. 4. Theoretcal Analyss We consder the pleentaton of the two partal decsons. Trgger partal decson s perfored only f an agent eets PLM and PDP. Once an agent perfors trgger partal decson, x 0 there wll be two cases. case. Recursve partal decson s recursvely perfored for tes ( 0) untl the neglected agent accepts the suggeston fro,.e., gu > 0. case 2. Recursve partal decson s recursvely perfored for tes ( 0) untl the neglected agent cannot prove ts x state by eans of recursve partal decson,.e., x g 0 To llustrate the effects of PDS, we assue that the orgnal local search has gotten stuc n a local nu before usng PDS, and wll not ncrease the ndvdual cost when usng PDS. Proposton When an agent x perfors trgger partal decson, the global cost s strctly decreasng n case. proof. We begn by ntroducng soe notatons. c (n), c (n), C (n) denote the local cost of x, the constrant cost between x and x, the global cost, respectvely, at the end of the n-th round. Assue that partal decson starts wth gnorng n the n-th round, x x 0 then receves the suggeston and gnores, and recursve partal decson s recursvely perfored for tes ( 0) untl gnores n the n+-th round, and accepts the suggeston fro x x x 2 x n the n++-th round. Accordngly, C (n-) and C (n++) are the global costs before and after the partal decson process, respectvely. We consder C (n-) - C (n++). C C ( n) ( n) [ c c... c c ] ( n) ( n) ( n) ( n) 0 pq N N N p0,,..., 0 0 q0,,..., [ c c... c c ( n) ( n) ( n) ( n) 0 pq N N N p0,,..., 0 0 q0,,..., [ c c... c c c ] ( n) ( n) ( n) ( n) ( n) 0 pq N N N N p0,,..., 0 0 q0,,..., [ c c... c c ( n) ( n2) ( n) 0 N N N 0 0 [ c c ] [ c c ]... [ ( n) ( n) ( n) ( n2) 0 0 N N N N c c ] [ c c ] ( n) ( n) ( n) ( n) N N N N [ c ( n+) pq p0,,..., p0,,..., q0,,..., q0,,..., c ( n) pq ] ]. (6) ( n) ( n) pq N p0,,..., q0,,..., c ] (7) (8) g g... g g 0 N, ( n) ( n) ( c ( x, ) ( *, )) v x v c x v x v (9) gu 0 g g... 0 g g u (20) Equaton (6) parttons the global cost nto +3 parts correspondng to the order of agents perforng partal decson. n nequaton (7), we transfors the suaton by c (+) c () whch s due to the fact that the agents can use local search to decrease ts cost. n equaton (8), we odfy the suaton accordng to the assocatve law of addton. n nequaton (9), we transfor the suaton by equaton (), (5) and the fact that the gan obtaned by an agent x between two rounds wll be no less than g by eans of the orgnal local search. The fnal equaton coes by equaton (2). Here, (0 -) and gu are greater than 0 because each neglected agent x assgns the suggested value v*. Thus, C (n++) s always less than C (n-) n case. Proposton 2 When an agent x perfors trgger partal decson, the local search wth PDS s equvalent to ts orgnal wth local restart n case 2. proof. Assue that partal decson starts by n-th round, then x g x 0 gnorng receves the suggeston and gnores x x 2 n the, and recursve partal decson s recursvely perfored for tes ( 0) untl cannot prove ts state by eans of partal decson (.e., + not change ts value, g + 0) n the n++-th round. Snce value. We consder C (n-) - C (n+). C C ( n) ( n) ( n) ( n) ( n) ( n2) 0 0 N N N N does s the last agent assgnng the suggested [ c c ] [ c c ]... [ c c ] [ c c ] 0 ( n2) ( n) ( n-) ( n) - - N N N N N, (2) ( n) ( n) c 0 c 0 0 N N 0 0 g g... 0 g g 2 ( n-) ( n) c ( x, ) ( *, ) v x v c x v x v ( n) ( n) c - c ( 0) 0 0 gu ( ) g g... ( 2) 0 g g 2 u (22) (23) nequaton (2), (22) and equaton (23) coe accordng to (8), (9) and (20). When = 0, s the only agent changng ts x 0 ( n) ( n) value after eetng PLM, so c -c 0 s less than 0. However, 0 when, g (0 -) s greater than 0 but gu s less than 0 accordng to Dsposng suggeston essage (lne 27-33). Consequently, whether C (n+) s less than C (n-) s unclear. 9

6 At ths pont, has gnored x but x suggested value. Thus, the current value of does not assgn the s not the best response n ters of the values of ts neghbors. So, n the next round, wll perfor the orgnal local search to change the current value, whch could nfluence ts neghbors, especally Slarly, when changes ts value, x orgnal local search to change ts value f x wll also perfor the x holds v * fro equaton (4). The above change wll happen recursvely n ore agents, whch s equvalent to a local search wth local restart.. Proposton 3. The convergence of PDS-based algorths wll eventually depend on ther orgnals. proof. The convergence of PDS-based algorths depends on two factors, the convergence of ther orgnal algorths and the effect of PDS. t can be seen fro forula (6) and (7) that PDP wll gradually approach to zero as C_round grows. Wth the decrease of PDP, trgger partal decson s less lely to be executed (lne 6). That s, the effect of PDS wll be gradually decreased durng the optzaton process. Accordngly, ore and ore agents wll perfor the orgnal local search (Lne 8-9). Fnally, PDSbased algorths wll perfor ust le ther orgnals. 5. Experental Analyss n order to deonstrate ts effect on dstrbuted local search, the partal decson schee (PDS) s appled to DSA (type-c and p=0.4), DSAN, MGM and MGM2 naed as PDS-DSA, PDS- DSAN, PDS-MGM and PDS-MGM2, respectvely. n our experents, we wll copare these PDS-based algorths wth ther orgnals and DSA-SDP on three dfferent types of probles: rando DCOPs, scale-free probles and graph-colorng probles. Here, DSA-SDP s reported the best ALS-DSA wth exploraton heurstcs n [7]. We average the experental results over 50 ndependently generated probles, each of whch s solved by each algorth 30 tes. The copared algorths except DSA- SDP are all pleented wthout the ALS fraewor. We consder rando DCOPs wth n = 20 agents, = 0 values n each doan, costs chosen fro the range, 2,..., 00, and constrant densty p = 0. (sparse probles) or p=0.6 (dense probles) [7]. For scale-free probles, we generate nstances by Barabáas-Albert odel [28] where an ntal set of 0=20 connected agents s used and n each teraton a new agent s connected to =3 other agents for sparse probles or =0 other agents for dense probles wth a probablty proportonal to the nuber of lns that the exstng agents already have. Besdes, the other paraeter settngs n scale-free networs are the sae as ones n rando DCOPs. For graph-colorng probles, we use n=20 agents, the nuber of colors=3 and the densty paraeter p=0.05 [7]. As n standard graph colorng probles, we set the cost of each volated constrant (two adacent varables wth the sae color) to one. These probles are nown to be hard Max- CSP probles,.e., beyond the phase transton between solvable and non-solvable probles [7]. Fgure 6. The cost n each step of all 9 algorths when solvng Rando DCOPs (sparse probles) Fgure 7. The cost n each step of all 9 algorths when solvng Rando DCOPs (dense probles) Fgure 6 and Fgure 7 show the coparson wth PDS-DSA, PDS-DSAN, PDS-MGM, PDS-MGM-2, ther orgnals and DSA- SDP on rando DCOPs wth p = 0. and p=0.6, respectvely. t can be seen that the PDS-based algorths have an obvous advantage over ther orgnals and DSA-SDP n sparse probles. The PDS-based algorths are superor to DSA-SDP by about 2.2%~4.0%. And the proveent of PDS-DSA over DSA s about 7.8%. PDS-DSAN proves DSAN by about 7.8%. And the proveent of PDS-MGM and PDS-MGM2 over ther orgnals are about.3% and 4.0%, respectvely. However, the proveents of the PDS-based algorths over ther orgnals are not obvous and only about.2%~4.5% n dense probles. Moreover, PDS-MGM2 and PDS-DSA s slghtly nferor to DSA-SDP. n addton, all the PDS-based algorths have advantages over ther orgnals at statstcally sgnfcant level of p-value < n sparse proble and p-value < n dense rando DCOPs, respectvely. Besdes, t can be found that the PDS-based algorths have the slar curves, especally for dense probles. The reason ght be that the PDS leads the local search process snce t has a better search ablty than the orgnal local search schees. Moreover, we fnd that PDS-MGM outperfors PDS-MGM2 whle MGM2 s better than MGM. The reason s that PDS-MGM has ore chances to perfor partal decson n the lted steps snce MGM2 and MGM requres 5 and 2 steps per round, respectvely. Here, the step refers to the councaton cycle between agents. 92

7 Fgure 8. The cost n each step of all 9 algorths when solvng Scale-Free probles (sparse probles) Fgure 9. The cost n each step of all 9 algorths when solvng Scale-Free probles (dense probles) Fgure 8 and Fgure 9 show the coparson wth all nne algorths for solvng sparse and dense scale-free probles, respectvely. Slar to rando DCOPs, the PDS-based algorths have great superorty over ther orgnals n sparse probles at statstcally sgnfcant level of p-value < The proveent of PDS-DSA over DSA s 4.7%. And PDS- DSAN proves the DSAN by about 2.8%. PDS-MGM and PDS-MGM2 also outperfor ther orgnals by 7.4% and 7.9%, respectvely. Besdes, t can be seen that the PDS-based algorths are superor to DSA-SDP by about 8.3%~9.2%. Moreover, the PDS-based algorths have stll advantages over ther orgnals by about 3.0%~7.4% n dense probles at statstcally sgnfcant level of p-value < Fgure 0 show the coparson wth all nne algorths for solvng graph colorng probles. We can see that DSAN and PDS-DSAN exhbt ore excellent perforance over the others. And, PDS-DSAN has the slar perforance wth DSAN, whch ndcates the local search schee adopted by DSAN s ore sutable for solvng DCSPs. Meanwhle, t can be observed that the other PDS-based algorths have proved ther orgnals a lot, especally for MGM. t llustrates that the PDS s also a good decson schee for DCSPs. 6. Concluson and Future Wor Each agent n the local search process needs to tae account of the current values of all ts neghbors so as to select ts value, whch would prohbt t fro searchng soe prosng solutons. Ths Fgure 0. The cost n each step of all 9 algorths when solvng graph colorng probles paper presents a Partal Decson Schee to brea the assgnent dependence. The PDS coprses trgger partal decson and recursve partal decson processes. The forer s teratvely perfored by agents who eet the trgger condtons, where they neglect soe bad values of ther neghbors to enhance ther local benefts and provde the suggested values for the neglected neghbors. The latter s recursvely perfored by neglected agents, where they gnore the values of ther neghbors except ther suggesters to prove global benefts. The partal decson schee can be appled to any local search algorth and the experental results verfy ts advantage on benchar probles. n the future, we wll probe nto new soluton evaluaton and control echans to enhance the convergence speed and accuracy of PDS-based local search algorths. n addton, we wll also try to extend the partal decson schee by gnorng the values of ultple neghbors. 7. REFERENCES [] Puol-Gonzalez, M. 20. Mult-agent coordnaton: Dcops and beyond. n: Proceedngs nternatonal Jont Conference on Artfcal ntellgence (JCA) (Vol. 22, No. 3, p. 2838). [2] Enebrec, F., Barthès, J. P. A Dstrbuted constrant optzaton wth MULBS: A case study on collaboratve eetng schedulng. Journal of Networ and Coputer Applcatons, 35(), [3] Sultan, E., Mod, P. J., Regl, W. C On Modelng Multagent Tas Schedulng as a Dstrbuted Constrant Optzaton Proble. n: Proceedngs nternatonal Jont Conference on Artfcal ntellgence (JCA). pp [4] Cheng, S., Raa, A., Xe, J Dynac ult-agent load balancng usng dstrbuted constrant optzaton technques. Web ntellgence and Agent Systes: An nternatonal Journal, 2(2), -38. [5] Farnell, A., Rogers, A., Jennngs, N. R Agent-based decentralsed coordnaton for sensor networs usng the ax-su algorth. n: Autonoous agents and ult-agent systes, 28(3), [6] Muldoon, C., O Hare, G. M., O Grady, M. J., Tynan, R., Trgon, N Dstrbuted constrant optzaton for resource lted sensor networs. n: Scence of Coputer Prograng, 78(5),

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