complete search would, in many cases, not be possible

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1 GENET: A Connectionist Achitectue fo Solving Constaint Satisfaction Poblems by Iteative Impovement Andew Davenpot, Edwad Tsang, Chang J. Wang and Kangmin Zhu Depatment of Compute Science, Univesity of Essex, Wivenhoe Pak, Colcheste, Essex CO4 3SQ, United Kingdom. fdaveat,edwad,cwang,kangming@essex.ac.uk Abstact New appoaches to solving constaint satisfaction poblems using iteative impovement techniques have been found to be successful on cetain, vey lage poblems such as the million queens. Howeve, on highly constained poblems it is possible fo these methods to get caught in local minima. In this pape we pesent genet, a connectionist achitectue fo solving binay and geneal constaint satisfaction poblems by iteative impovement. genet incopoates a leaning stategy to escape fom local minima. Although genet has been designed to be implemented on vlsi hadwae, we pesent empiical evidence to show that even when simulated on a single pocesso genet can outpefom existing iteative impovement techniques on had instances of cetain constaint satisfaction poblems. Intoduction Recently, new appoaches to solving constaint satisfaction poblems (csps) have been developed based upon iteative impovement (Minton et al. 1992; Selman & Kautz 1993; Sosic & Gu 1991). This technique involves st geneating an initial, possibly \awed" assignment of values to vaiables, then hill-climbing in the space of possible modications to these assignments to minimize the numbe of constaint violations. Iteative impovement techniques have been found to be vey successful on cetain kinds of poblems, fo instance the min-conicts hill-climbing (Minton et al. 1992) seach can solve the million queens poblem in seconds, while gsat can solve had, popositional satisability poblems much lage than those which can be solved by moe conventional seach methods. These methods do have a numbe of dawbacks. Fistly, many of them ae not complete. Howeve, the size of poblems we ae able to solve using iteative impovement techniques can so lage that to do a Andew Davenpot is suppoted by a Science and Engineeing Reseach Council Ph.D Studentship. This eseach has also been suppoted by a gant (GR/H75275) fom the Science and Engineeing Reseach Council. complete seach would, in many cases, not be possible anyway. A moe seious dawback to iteative impovement techniques is that they can easily get caught in local minima. This is most likely to occu when tying to solve highly constained poblems whee the numbe of solutions is elatively small. In this pape we pesent genet, a neual-netwok achitectue fo solving nite constaint satisfaction poblems. genet solves csps by iteative impovement and incopoates a leaning stategy to escape local minima. The design of genet was inspied by the heuistic epai method (Minton et al. 1992), which was itself based on a connectionist achitectue fo solving csps the Guaded Discete Stochastic (gds) netwok (Adof & Johnston 1990). Since genet is a connectionist achitectue it is capable of being fully paallelized. Indeed, genet has been designed specifically fo a vlsi implementation. Afte intoducing some teminology we descibe a genet model which has been shown to be eective fo solving binay csps (Wang & Tsang 1991). We intoduce extensions to this genet model to enable it to solve poblems with geneal constaints. We pesent expeimental esults compaing genet with existing iteative impovement techniques on had gaph coloing poblems, on andomly geneated geneal csps and on the Ca Sequencing Poblem (Dincbas, Simonis, & Van Hentenyck 1988). Finally, we biey explain what we expect to gain by using vlsi technology. Teminology We dene a constaint satisfaction poblem as a tiple (Z; D; C) (Tsang 1993), whee: Z is a nite set of vaiables, D is a function which maps evey vaiable in Z to a set of objects of abitay type. We denote by D x the set of objects mapped by D fom x, whee x 2 Z. We call the set D x the domain of x and the membes of D x possible values of x. C is a set of constaints. Each constaint in C esticts the values that can be assigned to the vaiables in Z simultaneously. A constaint is a nogood

2 if it fobids cetain values being assigned to vaiables simultaneously. An n-ay constaint applies to n vaiables. A binay csp is one with unay and binay constaints only. A geneal csp may have constaints on any numbe of vaiables. We dene a label, denoted by hx; vi, as a vaiablevalue pai which epesents the assignment of value v to vaiable x. A compound label is the simultaneous assignment of values to vaiables. We use (hx 1 ; v 1 i;:::;hx n ; v n i) to denote the compound label of assigning v 1 ; :::;v n to x 1 ; :::;x n espectively. A k-compound label assigns k values to k vaiables simultaneously. A solution tuple of a csp is a compound label fo all the vaiables in the csp which satises all the constaints. Binay GENET Netwok Achitectue The genet neual netwok achitectue is similia to that of the gds netwok. In the genet netwok each vaiable i in Z is epesented by a cluste of label nodes, one fo each value j in its domain. Each label node may be in one of two states \on" o \o". The state S hi;ji of a label node epesenting the label hi; ji indicates whethe the assignment of the value j to vaiable i is tue in the cuent netwok state. The output of a label node V hi;ji is 1ifS hi;ji is \on" and 0 othewise. All binay constaints in genet must be epesented by nogood gound tems. Binay constaints ae implemented as inhibitoy (negatively weighted) connections between label nodes which may be modied as a esult of leaning. Initially all weights ae set to 1. The input to each label node I hi;ji is the weighted sum of the output of all the connected label nodes: I hi;ji = X k2z;l2dk W hi;jihk;li V hk;li (1) whee W hi;jihk;li is the connection weight between the label nodes epesenting the labels hi; ji and hk; li 1. Since thee ae only connections between incompatible label nodes the input to a label node gives an indication of how much constaint violation would occu should the label node be in an on state. If no violation would occu the input would be a maximum of zeo. A csp is solved when the input to all the on label nodes is zeo such a state is called a global minima. Each cluste of label nodes is govened by a modulato which eectively implements a vaiation of the min-conicts heuistic (Minton et al. 1992). The pupose of the modulato is to detemine which label node in the cluste is to be on. Only one label node in a cluste may be on at any one time. The modulato selects the label node with the highest input to be on, with ties 1 If thee is no constaint between two label nodes epesenting hi; ji and hk; li then W hi;jihk;li = 0. being boken andomly. When the modulato changes the label node which is on in a cluste we say it has made a epai. GENET Convegence Pocedue A state of a genet netwok epesents a complete assignment of values to vaiables i.e. exactly one label node in each cluste is on. The initial state of the genet netwok is detemined andomly one label node pe cluste is selected abitaily to be on. genet iteates ove convegence cycles until it nds a global minima. We dene a convegence cycle as: 1. foeach cluste in paallel do 2 update states of all label nodes in cluste, 2. if none of the label nodes have changed state in step 1 then (a) if the input to all on nodes is zeo then solution found teminate, (b) else activate leaning, 3. goto step 1. Leaning Like most hill-climbing seaches, genet can each local optimal points in the seach space whee no moe impovements can be made to the cuent state in this case we say the netwok is in a minima. A local minima is a minima in which constaints ae violated. genet can sometimes escape such minima by making sideways moves to othe states of the same \cost". Howeve in some minima this is not possible, in which case we say the netwok is in a single-state minima. To escape local minima we adjust the weights on the connections between label nodes which violate a constaint accoding to the following ule: 3 W t+1 hi;jihk;li = W t hi;jihk;li V hi;jiv hk;li (2) whee W t is the connection weight between label hi;jihk;li nodes epesenting hi; ji and hk; li at time t. By using weights we associate with each constaint a cost of violating that constaint. We can also associate with each genet netwok state a cost which is the sum of the magnitudes of the weights of all the constaints violated in that state. Leaning has the eect of \lling in" local minima by inceasing the cost of violating the constaints which ae violated in the minima. Afte leaning, constaints which wee violated in the minima ae less likely to be violated again. This can be paticulaly useful in 2 We do not want clustes to update thei states at exactly the same time since this may cause the netwok to oscillate between a small numbe of states indenitely. In a vlsi implementation we would expect the clustes to update at slightly dieent times. 3 Mois (Mois 1993) has ecently epoted a similia mechanism fo escaping minima.

3 stuctued csps whee some constaints ae moe citical than othes (Selman & Kautz 1993). Leaning is activated when the genet netwok state emains unchanged afte a convegence cycle. Thus leaning may occu when genet, given the choice of a numbe of possible sideways moves to states of the same cost, makes a sideways move back to its cuent state. We conside this a useful featue of genet since it allows the netwok to escape moe complicated multi-state minima composed of a \plateau" of states of the same cost. A consequence of leaning is that we can show genet is not complete. This is because leaning aects many othe possible netwok states as well as those that compose the local minima. As a esult of leaning new local minima may be ceated. A discussion of the poblems this may cause can be found in (Mois 1993). Geneal Constaints Many eal-life csps have geneal constaints e.g. scheduling, ca sequencing (Dincbas, Simonis, & Van Hentenyck 1988). In this section we descibe how can we epesent two types of geneal constaint, the illegal constaint and the atmost constaint, in a genet netwok. One of ou motivations fo devising these paticula constaints has been the Ca Sequencing Poblem, a eal-life geneal csp once consideed intactable (Paello & Kabat 1986) and which has been successfully tackled using csp solving techniques (Dincbas, Simonis, & Van Hentenyck 1988). Since we cannot epesent geneal constaints by binay connections alone, we intoduce a new class of nodes called constaint nodes. A constaint node is connected to one o moe label nodes. Let c be a constaint node and L be the set of label nodes which ae connected to c. Then the input I c to the constaint node c is the unweighted sum of the outputs of these connected label nodes: I c = X hi;ji2l V hi;ji (3) We can conside the connection weights between constaint nodes and label nodes to be assymetic. The weight on all connections fom label nodes to constaint nodes is 1 and is not changed by leaning. Connection weights fom constaint nodes to label nodes ae, like fo binay constaints, initialised to 1 and can change as a esult of leaning. The input to label nodes in netwoks with geneal constaints C is now given by: I hi;ji = X W c;hi;ji V c;hi;ji X W hi;jihk;li V hk;li + k2z;l2dk c2c (4) whee V c;hi;ji is the output of the constaint node c to the label node hi; ji. The leaning mechanism fo connection weights W t between constaint nodes c and label nodes c;hi;ji hi; ji is given by: W t+1 W t = c;hi;ji 1 if S c > 0 c;hi;ji othewise W t c;hi;ji whee S c is the state of the constaint node. (5) The Illegal Constaint The illegal(hx 1 ; v 1 i;:::;hx k ; v k i) constaint species that the k-compound label L =(hx 1 ; v 1 i;:::;hx k ; v k i) is a nogood. An illegal constaint is epesented in a genet netwok by an illegal constaint node, which is connected to the k label nodes which epesent the k labels in L. S ill = I ill (k 1) (6) The state S ill of the illegal constaint node is negative if less than k 1 of connected label nodes ae on. In this case thee is no possibility that the constaint will become violated should anothe node become on. A constaint node in this state outputs 0 to all the connected label nodes. If k 1 of the connected label nodes ae on then we want to discouage the emaining o label node fom becoming on, since this will cause the constaint to be violated. Howeve, we do not wish to penalize the label nodes which ae aleady on, since the constaint emain satised even if they do change state. In this case we want to output 1 to the label node which is o and 0 to the emaining label nodes. Finally, if all the connected label nodes ae on then the constaint is violated. We want to penalize all these nodes fo violating the constaint, so we give them all an output of 1 to encouage them to change state. We summaize the output V ill;hi;ji fom an illegal constaint node ill to a label node epesenting the label hi; ji by: V ill;hi;ji = 0 if S ill < S ill V hi;ji othewise (7) The Atmost Constaint We can easily extend the illegal constaint node achitectue to epesent moe complex constaints. Fo instance, given a set of vaiables Va and values Val the atmost(n, Va, Val) constaint species that no moe than N vaiables fom Va may take values fom Val. The atmost constaint node is connected to all nodes of the set L which epesent the labels fhi; jiji 2 Va, j 2 Val, j 2 D i g. This constaint is a modication of the atmost constaint found in the chip constaint logic pogamming language. The state S atm of an atmost constaint node is detemined as follows: S atm = I atm N (8)

4 The output fom an atmost constaint node is simila to that fo the illegal constaint node, although we have the added complication that a single vaiable may have moe than one value in the constaint. We do not want label nodes in the same cluste to eceive dieent inputs fom a paticula constaint node since, in situations whee the netwok would nomally be in a single state local minima, we would nd the netwok oscillating about the states of these label nodes. Instead, we give the output of an atmost constaint node atm to a label node epesenting the label hi; ji as follows: ( 0 if S atm < 0 V atm;hi;j i = 1 MaxfV hi;ki jk 2 Val g if S atm = 0 1 othewise (9) Expeimental Results Gaph Coloing In (Selman & Kautz 1993) it is epoted that the pefomance of gsat on gaph coloing poblems is compaable with the pefomance of some of the best specialised gaph-coloing algoithms. This supised us since a gaph coloing poblem with N vetices to be coloed with k colos would equie, in a conjunctive nomal fom (cnf) epesentation, N k vaiables. Since each of these vaiables has a domain size of 2 the size of the seach space is 2 Nk. To epesent such a poblem as a csp would equie N vaiables of domain size k, giving a seach space of size k N. Fo example, the 250 vaiable 29 coloing poblem in Table 1 has a seach space size in genet of possible states. This is fa smalle than the coesponding size of states possible in gsat. Anothe dieence between gsat and genet is the way in which they make epais. gsat picks the best \global" epai which educes the numbe of conicts amongst all the vaiables, wheeas genet makes \local" epais which minimizes the numbe of conicts fo each vaiable. Thus we would expect epais made by gsat to be of \highe quality" than those of genet, although they ae made at the exta expense of consideing moe possibilities fo each epai. We compaed gsat 4 and genet 5 on a set of had gaph coloing poblems descibed in (Johnson et al. 1991), unning each method ten times on each poblem. We pesent the esults of ou expeiments in Tables 1 and 2. Both gsat and genet managed to solve all the poblems, although gsat makes many moe epais to solve each poblem. These esults seem to conm ou conjectue that fo csps such as gaph coloing genet is moe eective than gsat because of the way it epesents such poblems. 4 We an gsat with max-flips set to 10 the numbe of vaiables, and with aveaging in eset afte evey 25 ties. 5 All expeiments wee caied out using a genet simulato witten in C++ on a Sun Micosystems Spac Classic. gaph median median numbe nodes colos time of epais hous 65; 197; secs 65; secs 2; hous 7; 429; 308 Table 1: gsat on had gaph coloing poblems. gaph median median numbe nodes colos time of epais hous 1; 626; secs 7; secs hous 571; 748 Table 2: genet on had gaph coloing poblems. Random Geneal Constaint Satisfaction Poblems Thee ae two impotant dieences between a sequential implementation of genet and min-conicts hillclimbing (mchc). The st is ou leaning stategy fo escaping local minima. The second dieence is in choosing which vaiables to update. mchc selects andomly a vaiable to update fom the set of vaiables which ae cuently in conict with othe vaiables. In genet we andomly select vaiables to update fom the set of all vaiables, egadless of whethe they conict with any othe vaiables. Ou aim in this expeiment was to ty to detemine empiically what eect these individual modications to mchc was making to the eectiveness of its seach. We compaed genet with a basic min-conicts hillclimbing seach, a modied mchc (mchc2) and a modied vesion of genet (genet2). mchc2 andomly selects vaiables to update fom the set of all vaiables, not just those which ae in conict. mchc2 can also be egaded as a sequential vesion of genet without leaning. In genet2 vaiables ae only updated if they ae in conict with othe vaiables. We poduced a set of geneal csps with vaying numbes of the atmost(n,va,val) constaint, whee N = 3, jvaj = 5 and jvalj = 5. The poblems wee not guaanteed to be solvable. Each poblem had fty vaiables and a domain of ten values. The set of vaiables and values in each constaint wee geneated andomly. At each data-point we geneated ten poblems. We an each poblem ten times with genet, genet2, mchc and mchc2. We set a limit of ve hunded thousand epais fo each un, afte which failue was epoted if no solution had been found. Figue 1 shows that mchc2 solves moe poblems than mchc. This is to be expected since, because mchc2 can modify the values of vaiables which ae

5 % Succ. Runs genet genet2 mchc mchc Numbe of atmost constaints Figue 1: Compaison of pecentage of successful uns fo genet and min-conicts hill-climbing seaches on andomly geneated geneal constaint satisfaction poblems. not in conict, it is less likely to become tapped in local minima. The pefomance of genet2 shows that leaning is an even moe eective way of escaping local minima. Howeve Figue 1 shows that combining these two appoaches in genet is the most eective way of escaping minima fo this paticula poblem set. The Ca Sequencing Poblem We have been using the ca-sequencing poblem as a benchmak poblem duing the development of a genet model which would solve geneal csps. The ca-sequencing poblem is a eal-life geneal csp which is consideed paticulaly dicult due to the pesence of global atmost constaints. Fo a full desciption of the ca sequencing poblem see (Dincbas, Simonis, & Van Hentenyck 1988). We compaed genet with mchc, mchc2 and chip. chip is a constaint logic pogamming language which uses a complete seach based on fowad-checking and the fail-st pinciple to solve csps. In ou expeiments we used andomly geneated poblems of size 200 cas and utilisation pecentages in the ange 60% to 80%. At each utilisation pecentage we geneated ten poblems. The poblems all had 200 vaiables with domains vaying fom 17 to 28 values and appoximately 1000 atmost constaints of vaying aity. All the poblems wee guaanteed to be solvable. We an the genet, mchc and mchc2 ten times on each poblem, with a limit of one million epais fo each un, afte which failue was epoted. This limit coesponded to a unning time of appoximately 220 seconds at 60% utilisation up to 270 seconds at 80% utilisation. We used the method descibed in (Dincbas, Simonis, & Van Hentenyck 1988) to pogam the poblems in chip, which mchc mchc2 utilisa- % succ. median % succ. median tion % uns epais uns epais Table 3: A compaison of mchc and mchc2 on 200 ca sequencing poblems. genet genet3 utilisa- % succ. median % succ. median tion % uns epais uns epais Table 4: A compaison of genet and genet3 on 200 ca sequencing poblems. included adding edundant constaints to speed up the seach. With a time limit of one hou to solve each poblem chip managed to solve two poblems at 60% utilisation, one poblem at 65% utilisation, two poblems at 70% utilisation and one poblem at 75% utilisation. The esults fo min-conicts hill-climbing and genet on 200 ca sequencing poblems ae given in Tables 3 and 4. Fom Table 3 it can be seen that mchc2 is moe effective than mchc at solving the ca-sequencing poblem. Howeve the esults fo mchc2 and genet ae vey similia, indicating that leaning is having vey little o no eect in genet. This can be attibuted to the pesence of vey lage plateaus of states of the same cost in the seach space. Leaning is activated only when genet stays in the same state fo moe than one cycle, thus leaning is less likely to occu when these plateaus ae lage. To emedy this poblem we made a modication to genet to foce leaning to occu moe often. We dene the paamete p sw as the pobability that, in a given convegence cycle, genet may make sideways moves. Thus, fo each convegence cycle, genet may make sideways moves with pobability p sw, and may only make moves which decease the cost with pobability 1 p sw. Thus, if genet is in a state whee only sideways moves may be made then leaning will occu with a pobability of at least 1 p sw. The esults fo genet3 in Table 4, whee p sw is set to 0.75, shows that this modication signicantly impoves the pefomance of genet.

6 VLSI Implementation Although the esults pesented so fa have been obtained using a genet simulato on a single pocesso machine, it is the aim of ou poject to implement genet on vlsi chips. A full discussion of a vlsi implementation fo genet would be beyond the scope of this pape 6 so in this section we descibe what we expect to gain by using vlsi technology. A disadvantage of the min-conicts heuistic, as noted by Minton (Minton et al. 1992), is that the time taken to accomplish a epai gows with the size of the poblem. Fo a single-pocesso implementation of genet the cost of detemining fo a single vaiable the best value to take is popotional to the numbe of values in the domain of the vaiable and the numbe constaints involving that value. To detemine fo each vaiable the best value to take can potentially be pefomed at constant time in a vlsi implementation of genet no matte how lage the domain o how highly constained the poblem. This would mean that the time taken fo genet to pefom a single convegence cycle would be constant, no matte what the poblem chaacteistics 7. Since we estimate the time taken to pefom one convegence cycle using cuent vlsi technology to be of the ode of tens of nanoseconds, this would allow all the csps mentioned in this pape to be solved in seconds athe than minutes o hous. Conclusion We have pesented genet, a connectionist achitectue fo solving constaint satisfaction poblems by iteative impovement. genet has been designed to be implemented on vlsi hadwae. Howeve we have pesented empiical evidence to show that even when simulated on a single pocesso genet can outpefom existing iteative impovement techiques on had binay and geneal csps, We have developed stategies fo escaping local minima which we believe signicantly extend the scope of hill-climbing seaches based on the min-conicts heuistic. We have pesented empiical evidence to show that genet can eectively escape local minima when solving a ange of highly constained, eal-life and andomly geneated poblems. Refeences Adof, H., and Johnston, M A discete stochastic neual netwok algoithm fo constaint satisfaction poblems. In Poceedings of the Intenational Joint Confeence on Neual Netwoks. Dincbas, M.; Simonis, H.; and Van Hentenyck, P Solving the ca-sequencing poblem in logic pogamming. In Poceedings of ECAI-88. Johnson, D.; Aagon, C.; McGeoch, L.; and Schevon, C Optimization by simulated annealing: an expeimental evaluation; pat II, gaph coloing and numbe patitioning. Opeations Reseach 39(3):378{ 406. Minton, S.; Johnston, M.; Philips, A.; and Laid, P Minimizing conicts: a heuistic epai method fo constaint satisfaction and scheduling poblems. Aticial Intelligence 58:161{205. Mois, P The beakout method fo escaping fom local minima. In Poceedings of the Twelth National Confeence on Aticial Intelligence. AAAI Pess/The MIT Pess. Paello, B., and Kabat, W. C Job-shop scheduling using automated easoning: A case study of the ca-sequencing poblem. JOURNAL of Automated Reasoning 2:1{42. Selman, B., and Kautz, H Domain independent extensions to GSAT: Solving lage stuctued satisability poblems. In Poceedings of the 13th Intenational Joint Confeence on Aticial Intelligence. Sosic, R., and Gu, J ,000,000 queens in less than one minute. SIGART Bulletin 2(2):22{24. Tsang, E Foundations of Constaint Satisfaction. Academic Pess. Wang, C., and Tsang, E Solving constaint satisfaction poblems using neual-netwoks. In Poceedings IEE Second Intenational Confeence on Aticial Neual Netwoks. Wang, C., and Tsang, E A cascadable VLSI design fo GENET. In Intenational Wokshop on VLSI fo Neual Netwoks and Aticial Intelligence. Acknowledgements We would also like to thank Alvin Kwan fo his useful comments on ealie dafts of this pape. We ae gateful to Bat Selman and Heny Kautz fo making thei implementation of gsat available to us. 6 A vlsi design fo genet is descibed in (Wang & Tsang 1992) 7 The size of poblem would be limited by cuent vlsi technology