An Algebraic Approach to Constraint Satisfaction Problems

Transcription

1 An Algebrac Approach to Constrant Satsfacton Problems Igor Rvn Wolfram Research, Inc. Champagn, Illnos Ramn Zabh Computer Scence Department Stanford Unversty Stanford, Calforna Abstract A constrant satsfacton problem, or CSP, can be reformulated as an nteger lnear programmng problem. The reformulated problem can be solved va polynomal multplcaton. If the CSP has n varables whose doman sze s m, and f the equvalent programmng problem nvolves M equatons, then the number of solutons can be determned n tme Onm2 M n ). Ths surprsng lnk between search problems and algebrac technques allows us to show mproved bounds for several constrant satsfacton problems, ncludng new smply exponental bounds for determnng the number of solutons to the n-queens problem. We also address the problem of mnmzng M for a partcular CSP. 1 Introducton We are nterested n solvng certan search problems that arse n combnatorcs and n artfcal ntellgence. These problems are called Constrant Satsfacton Problems CSP s), and nclude such famlar tasks as graphcolorng and the n-queens problem. Typcally, such problems are solved va exhaustve search. We propose an alternatve method, based on convertng the CSP nto an nteger lnear programmng problem, whch can be solved va polynomal multplcaton. In secton 2 we descrbe constrant satsfacton problems and show how they can be converted nto nteger lnear programmng problems. We descrbe how to solve nteger lnear programmng problems va polynomal multplcaton n secton 3, and analyze the tme requrements of ths approach. In secton 4 we gve an example of our method, applyng t to the n-rooks problem, and show some surprsng smlartes wth standard search technques. In secton 5 we address the problem of fndng a good nteger lnear programmng equvalent of a CSP, whle n secton 6 we descrbe some extensons to our technques. Fnally, n secton 7, we derve new smply exponental bounds for the n-queens problem and the torodal n-queens problem. 2 CSP s and nteger lnear programmng 2.1 Constrant satsfacton problems Formally, a CSP has a set of varables and a doman of values, V = {v 1,v 2,...,v n } the set of varables, D = {d 1,d 2,...,d m } the set of values. Every varable x must be assgned a value d j ; such an assgnment wll be denoted v d j. A CSP also conssts of some constrants sayng whch assgnments of values to varables are compatble. We wll restrct our attenton to bnary CSP s, where the constrants nvolve only two varables. Ths s partly for smplcty, and partly because a non-bnary CSP can be converted nto a bnary CSP by the ntroducton of addtonal varables.) A soluton to the constrant satsfacton problem s an n-tuple of assgnments that satsfes all the constrants. There are several ways to formulate the task of solvng a CSP. The satsfablty problem s to determne the exstence of a soluton to a CSP. The countng problem s to determne the number of solutons. The enumeraton problem s to fnd all solutons. The countng problem s a specal case of the enumeraton problem, and the satsfablty problem s a specal case of the countng problem. Snce many NP-complete problems, such as graphcolorng, are CSP s, the satsfablty problem s easly seen to be NP-complete [Garey and Johnson, 1979]. We therefore cannot expect to solve the satsfablty, countng or enumeraton problems n polynomal tme. Typcal CSP soluton technques lke those surveyed n [Mackworth, 1987] perform an exhaustve search, backtrackng

2 through all n-tuples of assgnments. Whle these methods add some addtonal cleverness beyond brute-force search, ther worst-case runnng tme s stll Om n ). We wll concentrate on solvng the countng problem and therefore the satsfablty problem as well). Whle our methods are easly extended to the enumeraton problem see secton 6.1), our approach to the countng problem s surprsngly smple to analyze. In fact, we can do provably better than backtrack search for some problems, n the followng sense. Backtrackng counts the number of solutons by enumeratng them, and hence must take at least as much tme as the number of solutons. Our technques can determne the number of solutons wthout actually enumeratng them, so the number of solutons s not a lower bound on our runnng tme. 2.2 Convertng CSP s nto nteger lnear programmng problems Our basc approach s to turn a CSP nto an nteger lnear programmng problem. For every possble assgnment v d j we ntroduce a varable x,j. Then for every CSP varable v we have an equaton 1 j m x,j =1. 1) Ths expresses the fact that every CSP varable has exactly one value. In addton, for every set of assgnments that the constrants prohbt, we have an nequalty whch states that no more than one of these assgnments can hold at once. For example, f the par of assgnments {v d j,v k d l } s forbdden by the constrants, we wll have the nequalty x,j + x k,l 1. 2) Now consder the problem of fndng all solutons to the equatons of the form 1) and 2) over the non-negatve ntegers. It s clear that any soluton wll assgn every varable x,j ether 0 or 1.) Ths s an nteger lnear programmng problem, whch corresponds to the orgnal CSP. There are nm varables x,j, and a soluton to the set of equatons wth x,j = 1 corresponds to a soluton to the CSP contanng the assgnment v d j. 3 Solvng CSP s by multplyng polynomals An nteger lnear programmng problem s a set of lnear Dophantne nequaltes. We can use an approach based on generatng functons to determne the number of solutons. Ths nvolves multplyng out a certan polynomal, and gves us a way of solvng the orgnal CSP. 3.1 Solvng lnear Dophantne equatons A set of M lnear Dophantne equaton n N unknowns has the form N w,j x j = s, =1,...,M. 3) The x j s are the unknowns, and the w,j s and s s are non-negatve ntegers. The problem at hand s to determne the number of solutons over the non-negatve ntegers. Consder the generatng functon ΦY 1,...,Y M )= N 1 1 ). 4) M =1 Y w,j The formal parameters of ths functon are the varables Y 1,...,Y M, and on such varable s assocated wth each equaton. We need one non-trval bt of mathematcs: the number of solutons to equaton 3) s the coeffcent of M Y s 5) =1 n Φ. Ths fact was proved n [Euler, 1748], and s dscussed n [Schrjver, 1986, pages ].) 3.2 Applcatons to nteger lnear programmng Now consder the nteger lnear programmng problem correspondng to a CSP. Every w,j wll be ether 0 or 1. Assume for the moment that all the equatons are of the form 1), hence s = 1. We wll relax ths restrcton shortly.) We can count the number of solutons by fndng the coeffcent of the monomal Y = Y 1 Y 2 Y M n Φ. We apply the dentty 1 1 x = k=0 x k to equaton 4), gvng N M ΦY 1,...,Y M )= 1+ k=1 =1 Y w,j ) k. 6) We can further smplfy ths f we remember that we are only nterested n the coeffcent of Y. We can therefore gnore any term dvsble by the square of any Y terms that are not so dvsble are called square free). By a slght abuse of notaton we wll refer to the square free part of the generatng functon as Φ. We next note that M M k=1 =1 Y w,j ) k = =1 Y w,j + other terms,

3 where these latter terms are not square free and so can be dscarded. Ths gves us an expresson for Φ as a product of bnomals, namely ) N M ΦY 1,...,Y M )= 1+. 7) =1 Y w,j All we need do n order to fnd the number of solutons to the system of equatons s compute the coeffcent of Y n equaton 7). The obvous way to solve ths problem s to smply multply out Φ and look at the correct term of the result. We can do ths teratvely: we multply the frst two bnomals of Φ and throw away the terms that are not square free, then repeat wth the next bnomal, etc. We can now relax our assumpton that all the equatons are of the form 1). Suppose that nstead we have exactly one Dophantne nequalty x,j 1. 8) 1 j m Ths mplctly defnes two equaltes, one of whch has 0 on the rght hand sde, and one of whch has 1. The number of solutons to the nequalty s the sum of the number of solutons to the two mplct equaltes. At frst glance, t would seem that solvng a system wth α nequaltes would requre solvng 2 α mplct systems of equaltes. However, a closer look at the generatng functon shows that we only need to solve one of these mplct systems. Suppose that we replace each nequalty wth an equalty that has a 1 on the rght hand sde and examne the square free part of Φ. The number of solutons to each of the 2 α mplct systems of equaltes wll be the coeffcent of the approprate term of Φ. For example, suppose that there are three equatons Y 0, Y 1 and Y 2, and one nequalty ϕ 1, 9) where ϕ s a sum. The nequalty 9) mplctly asserts ϕ =0 9.0) or ϕ =1. 9.1) The number of solutons to the orgnal system whch ncluded 9)) s the sum of the number of solutons when 9) s replaced by 9.0) plus the number of solutons when 9) s replaced by 9.1). However, suppose that we smply replace 9) by 9.1) and calculate Φ. Let Y 3 be the generatng functon varable assocated wth 9.1). The number of solutons s the coeffcent of Y 0 Y 1 Y 2 Y 3 n Φ. However, the number of solutons when 9) s replaced by 9.0) s also part of Φ, namely the coeffcent of Y 0 Y 1 Y 2. We thus do not need to calculate separate generatng functons for both systems of equaltes mplctly defned by 9). We can smply replace the nequalty by an equalty wth 1 on the rght hand sde and calculate Φ. At the end, we add together coeffcents for the mplct systems of equaltes to determne the number of solutons to the orgnal problem. To summarze, we can replace the α nequaltes wth equaltes and calculate the square free part of Φ. The total number of solutons wll no longer be the coeffcent of Y, but rather the sum of 2 α coeffcents of dfferent terms. Ths justfes our earler assumpton that the nteger lnear programmng problem conssted entrely of equaltes of the form 1). 3.3 Tme requrements The tme requred to solve the CSP s the tme to multply out the polynomal Φ. We can do ths n N teratons, where on each teraton we multply a bnomal by a polynomal that represents the product thus far. Two polynomals can be multpled n tme proportonal to the product of ther sze, so we need to bound the sze of the ntermedate results. Navely, there could be as many as N M terms, snce there are MY s, and the degree of each one s potentally as hgh as N. However, remember that at each stage we wll dscard all the terms that are not square free. In algebrac termnology, ths s descrbed as dong the computaton over the polynomal rng Z[Y 1,...,Y M ]/Y1 2,...,Y2 M ) nstead of Z[Y 1,...,Y M ].) Ths gves a better bound on the number of terms. Every Y wll have degree 0 or 1, hence there are no more than 2 M terms. Summarzng, f we compute Φ teratvely we wll do N multplcatons, where each multplcaton computes the product of a bnomal and a polynomal wth no more than 2 M terms. Each multplcaton wll take tme 2 M+1, whch s O2 M ). Ths puts a bound of ON2 M ) on the total tme complexty of our algorthm. 1 We wll show n secton 6.2 how to reduce the exponent to M n n the worst case. It s worth notng that we cannot expect to fnd a polynomal tme method for determnng the value of an arbtrary coeffcent of Φ. [Schrjver, 1986] proved that nteger lnear programmng s NP-complete even when the coeffcents nvolved are restrcted to be 0 or 1. 4 An example of our approach The n-rooks problem conssts of placng n rooks on an n-by-n chessboard so no two rooks attack each other. 1 Ths analyss gnores the tme requred for scalar multplcaton. We can take ths nto account by notng that the sze of the coeffcents wll be bounded by M N, hence ther length s ON log M). So the scalar multplcaton can be done n tme ON log M logn log M) log logn log M)) [Aho et al., 1974].

4 Whle t s clear that the solutons to ths problem are the n! permutatons of n elements, ths problem has a partcularly smple encodng as an nteger lnear programmng problem, and s therefore a useful example of our approach. There are n 2 assgnments v d j for ths CSP, so we wll convert t nto a programmng problem wth n 2 varables x,j. There wll be 2n equatons. 2 The equatons specfy that there s exactly one rook n each row and each column. A typcal equaton s x 2,1 + x 2, x 2,n =1, 10) whch encodes the fact that there s exactly one rook n the 2nd column. We can number these 2n equatons so that equaton says that there s exactly one rook n the th column, and equaton n+ says that there s exactly one rook n the th row. The formal parameter Y of the generatng functon corresponds to the th constrant equaton. Applyng equaton 7), Φ s a product of n 2 bnomals. There wll be one bnomal for every square on the chessboard, and the bnomal for the square n row and column j wll be 1 + Y n+ Y j ). The number of solutons s the coeffcent of Y = Y 1 Y n Y n+1 Y 2n n the expanson of Φ. Surprsngly, there s a way of multplyng out Φ that corresponds closely to backtrack search. Suppose that we attempt to fnd all combnatons of bnomal terms whose product s Y. We can start by pckng the second.e., non-constant) term of 1 + Y n+ Y j ). Ths forces us to pck the frst constant) term of every bnomal whose second term ncludes ether Y n+ or Y j, to ensure the product s square free. Now we can pck the non-constant term of some other bnomal and contnue, untl we have produced Y. The number of such combnatons wll be the coeffcent of Y n the expanson of Φ. It s also possble that the combnaton of terms selected guarantees that no selecton of terms from the remanng bnomals wll produce Y. The smplest example of ths occurs when a term has been selected from all but one bnomal wthout producng Y, where the remanng bnomal s non-constant term does not contan the mssng Y. Ths phenomenon, whch does not occur n the n-rooks problem, wll be used to mprove our performance guarantee n secton 6.2.) Ths whole process corresponds closely to a smple search of the orgnal CSP. Choosng the non-constant term of the bnomal 1 + Y n+ Y j ) places a rook on the equvalent square, whle choosng the constant term means there s no rook on that square. The process 2 In secton 5 we wll dscuss how to fnd good encodngs. descrbed above corresponds to placng a rook on the chessboard square n row and column j, and thus ensurng there s no rook n any other square n that row or column. At the end, we wll have satsfed every constrant and produced a soluton. The alternatve outcome, where we cannot satsfy every constrant, corresponds to a falure n backtrack search. Note that the expanson of Φ does not explctly represent the solutons. In fact, whch assgnments consttute a partcular soluton s lost by ths approach, as the dentty of the bnomals s not recorded. We wll show how to explctly represent solutons n secton 6.1. However, the fact that solutons are not explctly represented s an advantage of our approach, as the number of solutons s no longer a lower bound to our runnng tme. Applyng the analyss of secton 3.3, for ths problem we have N = n 2 and M =2n, so our runnng tme s On 2 4 n ). Ths s sgnfcantly smaller than n!, the number of solutons, whch s about 4 On log n). We have just descrbed a correspondence between a way to compute the coeffcent of Y n the expanson of Φ and a smple search technque for solvng the orgnal CSP. However, searchng all the combnatons of bnomal terms s not our ntended method. We beleve t would be better to multply Φ out teratvely, as descrbed earler. Whle ths does not correspond n any obvous way to applyng backtrack search to the orgnal problem, t does seem to have some smlarty to breadth-frst technques. It may thus be susceptble to parallel mplementaton technques lke those descrbed n [Dxon and de Kleer, 1988]. 5 Fndng a good encodng A potental dffculty wth our approach s that the runnng tme s exponental n M, the number of equatons needed to encode the CSP. There can be many such encodngs, and t s mportant to be able to fnd a good one. Clearly M must be at least n, as there s an equaton of the form 1) for each CSP varable. The remanng equatons come from the constrants. The problem s to fnd a way of encodng the constrants as small number of equatons. In the smplest encodng, there s an equaton lke 9) for every par of assgnments that the constrants rule out. However, ths can be qute a lot of equatons, as the followng example llustrates. In the case where all pars of assgnments are ruled out ths wll result n n 2 m 2 equatons, whle, as we wll see, there s an encodng that results n only 1 equaton. Let us defne the assgnment graph correspondng to a partcular CSP. The nodes of the assgnment graph represent assgnments, and there s an edge between two nodes f ther assgnments are ruled out by the constrants. A clque C n ths graph corresponds to a set of parwse ncompatble assgnments. The transformaton descrbed n secton 2.2 wll produce a varable x,j for

5 every node, and wll use n 2) equatons, wth the equaton x,j + x k,l 1 expressng the fact that these two assgnments are ncompatble. However, we can descrbe the fact that all the assgnments n C are parwse ncompatble wth the sngle equaton x,j + x k,l }{{} C So we really only need one equaton for every clque. If all pars of assgnments are ncompatble, the entre assgnment graph s a clque, and so the constrants can be represented wth 1 equaton nstead of needng n 2 m 2. For the n-queens problem, the assgnment graph has a node for each square on the chessboard. Two nodes are connected f queens on the two correspondng squares would attack each other. The maxmal clques correspond to the columns, rows, dagonals and antdagonals of the chessboard. We can therefore reduce the sze of M by encodng every clque as a sngle equaton. In an optmal encodng, M can be as small as the sze of the smallest coverng by clques of the assgnment graph. Because the problem of coverng a graph by clques s NP-complete [Garey and Johnson, 1979], fndng an optmal encodng would take exponental tme. For specal cases lke the chessboard problems descrbed above, t s easy to fnd an optmal encodng, but n general we would have to settle for a sub-optmal encodng. It s also possble to phrase the problem as fndng a small vertex cover for the assgnment graph, and ths can also reduce M. 5.1 Whch CSP s have good encodngs? One obvous queston s whether there s a better characterzaton of the CSP s that have good encodngs. Whle we do not have a very good answer to ths queston, an observaton due to Bob Floyd suggests that there may be a relatonshp between the number of solutons to the CSP and the smallest possble value for M. Floyd has ponted out that a soluton to the CSP corresponds to an n-clque n the complement of the assgnment graph equvalently, an ndependent set of sze n n the assgnment graph). It s possble that work n combnatorcs wll produce a relatonshp between the number of ndependent sets of fxed sze n a graph and the sze of the smallest vertex cover. Such a relatonshp would produce a characterzaton of the CSP s where our algorthm can be expected to work well, n terms of the number of solutons. 6 Some extensons The method descrbed above reles on multplyng out the polynomal Φ defned n equaton 7). A smple modfcaton to Φ allows us to solve the enumeraton problem nstead of the countng problem. It s also possble to reduce the runnng tme of our algorthm by carefully arrangng the order n whch the bnomals comprsng Φ are multpled out. 6.1 Solvng the enumeraton problem Suppose that nstead of determnng the number of solutons to our CSP, we are nterested n enumeratng these solutons. We can stll convert the CSP nto an nteger lnear programmng problem, and solve t va polynomal multplcaton. The only dfference s that we now need to consder a slghtly dfferent functon than Φ. Recall that we start by convertng our CSP nto a set of equatons n N = mn varables, one varable per assgnment v d j. Let us ntroduce N dfferent varables α j,1 j N. Now consder a slght varaton of Φ, namely ) N M Φ 1 = 1+α j. 11) =1 Y w,j We can solve the enumeraton problem by determnng the coeffcent of Y n the expanson of Φ 1, just as we dd wth Φ. Suppose we multply out Φ 1 by selectng terms from dfferent bnomals. The α j s wll keep track of whch bnomals had ther non-constant sde selected. When we have a soluton, the coeffcent of Y wll the product of a number of the α j s, whch wll encode a soluton. When Φ 1 s multpled out completely teratvely or otherwse), the coeffcent of Y wll not be an nteger, but rather the sum of a number of dfferent products of α j s. Every such product s a soluton to the CSP, and all solutons appear as such products. 6.2 Reducng the runnng tme An observaton due to John Lampng can be used to reduce the runnng tme of our algorthm from Onm2 M ) to Onm2 M n ). Suppose that before we multply out Φ we frst try to dentfy a subset S of the bnomals comprsng Φ and a varable Y such that Y never appears outsde of the bnomals n S, and some other varable Y does not appear n S. If we had such an S and Y, we could effectvely reduce the number of varables by 1. We can do ths by frst computng the product of the bnomals n S. Ths can be grouped nto terms based on the hghest power of Y that dvdes each term S = S 0 + YS 1 + Y 2 S 2... where S s not dvsble by Y, and as usual we can dscard the non square free terms. If Y s contaned n the coeffcents) correspondng to solutons to the CSP, then we can replace S by S 1. We can then forget about Y,as t never appears n the remanng part of Φ. Snce Y dd

6 not appear n S, ths reduces the number of varables by 1. Reducng the number of varables reduces the runnng tme of our algorthm as well, whch s exponental n the number of varables. It turns out that we are guaranteed that there exst at least n such pars of S s and Y s. Ths enables us to reduce the number of varables from M to M n, whch n turn reduces our runnng tme to Onm2 M n ). There are n such S and Y pars because the encodng of a CSP wll have a sngle equaton for each CSP varable, representng the fact that the varable must be assgned exactly one value. 3 Let Y correspond to the equaton whch constrans the CSP varable v to have exactly one value, and consder the bnomals n Φ that contan Y. These are exactly the bnomals that correspond to possble values for v, so there can be only m of them. Furthermore, ths set of bnomals forms an approprate S, snce no Y correspondng to another CSP varable can appear. There are n such Y s, one per CSP varable, and these Y s wll always appear n the coeffcents) that we need to calculate. It s even possble to take advantage of such S s and Y s when Y does not necessarly appear n the requred coeffcents. Recall that ths s a result of the nteger lnear programmng equaton that corresponds to Y actually representng an nequalty, as dscussed at the end of secton 3.2.) If ths happens, we can replace S by S 0 + S 1 and stll dscard Y. Ths has the effect of summng the coeffcents that stand for the solutons to the two mplct systems of equaltes early n the computaton, rather than watng to multply Φ out before calculatng ths sum. We can thus reduce the number of varables by at least n. As we saw n secton 4, each bnomal corresponds to a possble assgnment v d j. The orderng that s guaranteed to reduce the number of varables by n corresponds to consderng all the assgnments for a fxed v before movng on to the next CSP varable. On the n-queens problem ths s equvalent to multplyng out the bnomals n row-major order. 7 Some new bounds An addtonal advantage of our approach s that we can produce non-trval upper bounds for solvng certan constrant satsfacton problems. So far, we have confned out attenton to CSP s based on placng peces on a chessboard. The classc such problem s the n-queens problem, whch was known to Gauss. The n-queens problem conssts of placng n queens on an n-by-n chessboard so that no two queens attack. A varaton of ths problem s the torodal n-queens problem, where lnes of attack between 3 An encodng that does not have ths property corresponds to a CSP whose unsatsfablty can be easly detected va arc consstency [Mackworth, 1977]. queens are consdered to wrap as f the board were a torus. The torodal n-queens problem has strctly fewer solutons than the regular) n-queens problem. To our knowledge there s no known bound for CSP search algorthms for ths problem, beyond the trval bound of n n. In fact, there s some evdence that the number of solutons to the torodal problem and hence to the regular problem) s larger than exponental [Rvn and Vard, 1989]. The torodal n-queens problem can be formulated as a smple set of equatons. There are n 2 varables x j. There are also 4n equatons, one for each column, row, dagonal top to bottom) and ant-dagonal. So we have M =4n, N = n 2. Applyng the above results, our algorthm can determne the number of solutons n tme On 2 8 n ). For the regular n-queens problem, agan N = n 2, but ths tme there are 22n 1) nequaltes for the dagonals. Ths gves us M =6n 2, and a runnng tme of On 2 32 n ). However, f we do the multplcaton n row-major order, t turns out that only 3n varables wll need to be actve at once. Ths can be seen by notng that the top row nvolves 3n constrants. Movng to the next row adds a new row constrant and removes the old row constrant. The far left square of the new row elmnates an antdagonal constrant all the squares nvolved n that constrant have been elmnated), and adds a new dagonal constrant, whle the far rght square elmnates a dagonal constrant and adds a new antdagonal constrant. Ths gves us a much better bound of On 2 8 n ). Our prelmnary experments wth ths algorthm have been promsng. 8 Conclusons We have shown a surprsng connecton between solvng search problems and multplyng certan polynomals. The performance s very smple to analyze, and we can show new bounds for several constrant satsfacton problems. Our approach also holds our the possblty of applyng sophstcated algebrac technques to constrant satsfacton problems. For example, t s possble that the coeffcent of Y n Φ can be estmated, thus provdng an approxmate count of the number of solutons. Also, complex mathematcal results lke those n [Anshel and Goldfeld, 1988] can be used to provde an upper bound on the number of solutons by studyng the generatng functon drectly, usng the theory of modular forms. Our algorthm also shares some surprsng propertes wth Sedel s method [Sedel, 1981]. Lke Sedel, we produce a method whch s easy to analyze, and whch gves a bound that s exponental n a certan parameter of the problem. Sedel s parameter f s a property of the topology of the constrant graph, whch can be shown to be O n) for planar graphs. Our parameter M s a property of the way the problem gets encoded as an n-

7 teger lnear programmng problem, whch can be shown to be lnear for varants of the n-queens problem. Sedel faces the problem of makng f small, whch n general requres exponental tme. Smlarly, we face the problem of makng M small, whch requres solvng an NP-hard problem nvolvng graph clques. The smlartes between our algorthm and Sedel s, as well as Lampng s observaton descrbed n secton 6.2, suggest that our algorthm may be vewed as dynamc programmng. We are currently explorng ths possblty [Rvn and Zabh, 1989]. [Schrjver, 1986] Alexander Schrjver. Theory of Lnear and Integer Programmng. John Wley and Sons, [Sedel, 1981] Ramund Sedel. A new method for solvng constrant satsfacton problems. In Proceedngs of IJCAI-81, Vancouver, BC, pages , August Acknowledgements We are ndebted to John Lampng and Ilan Vard for useful suggestons. Support for ths research was provded by DARPA under contract number N C Ramn Zabh s supported by a fellowshp from the Fanne and John Hertz foundaton. References [Aho et al., 1974] Alfred Aho, John Hopcroft, and Jeffrey Ullman. The Desgn and Analyss of Computer Algorthms. Addson-Wesley, [Anshel and Goldfeld, 1988] Mchael Anshel and Doran Goldfeld. Applcatons of the Hardy-Ramanujan partton theory to lnear Dophantne equatons. Unpublshed pre-prnt, [Dxon and de Kleer, 1988] Mke Dxon and Johan de Kleer. Massvely parallel assumpton-based truth mantenance. In Proceedngs of AAAI-88, St. Paul, MN, pages Amercan Assocaton for Artfcal Intellgence, Morgan Kaufmann, August [Euler, 1748] L. Euler. Introducto Analysn Infnstorum, vol. 1. M. M Bousquet, Lausanne 1748; German translaton by H. Maser: Enletung n de Analyss des Unendlchen Erster Tel, Sprnger, Berln [Garey and Johnson, 1979] Mchael Garey and Davd Johnson. Computers and Intractablty. W. H. Freeman and Company, [Mackworth, 1977] Alan Mackworth. Consstency n networks of relatons. Artfcal Intellgence, 8:99 118, [Mackworth, 1987] Alan Mackworth. Constrant satsfacton. In Stuart Shapro, edtor, Encylopeda of Artfcal Intellgence. Wley-Interscence, [Rvn and Vard, 1989] Igor Rvn and Ilan Vard. The n-queens problem. Forthcomng. [Rvn and Zabh, 1989] Igor Rvn and Ramn Zabh. An algebrac approach to constrant satsfacton problems. Forthcomng extended verson.