Abstraction and the CSP Phase Transition Boundary Robert Schrag and Daniel Miranker
Department of Computer Sciences and The Applied Research Laboratories Taylor Hall 2.124, Mail Code C0500 University of Texas at Austin Austin, TX USA 78712 Internet: fschrag,
[email protected] WWW: http://www.cs.utexas.edu/users/schrag/ March 14, 1996 Abstract Domain abstraction is a ecient method for solving CSPs which is sound and incomplete with respect to unsatis ability | it is e ective only when both the input problem instance and the reduced instance are unsatis able. We characterize the e ectiveness of domain abstraction with reference to a random problem space which exhibits a sharp surface of transition in average frequency of satis ability (\phase transition"), delineating a boundary for the e ectiveness of domain abstraction. The abstraction has the e ect of loosening constraints, making the reduced instances more satis able on average. This loosening is accelerated as the degree of abstraction is increased, so that e ectiveness drops o suddenly, as another phase transition. We investigate alternative abstraction mapping policies, and we conjecture that the same pattern of sudden degradation will occur for any such policy. Qualitatively, the satis ability phase transition and constraint loosening restrict the e ectiveness of domain abstraction to input CSPs with very tight constraints. Quantitatively, we develop a series of analytical approximations to predict the location of the phase transition for abstraction
Corresponding author.
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e ectiveness, and we present empirical evidence verifying the approximations. Of possible independent interest, our analysis includes corrections to a standard formula approximating the location of the phase transition surface which are important for the case of CSP ensembles with very sparse constraint graphs.
1 Introduction Consider the available methods for addressing the constraint satisfaction problem (CSP) which are sound but sacri ce completeness to gain eciency. There exists a number of stochastic constraint satisfaction methods which are sound and incomplete with respect to satis ability | any solution they return is a valid solution, but they may not nd a solution when it exists (e.g., [Selman et al., 1992, Davenport et al., 1994]). These methods frequently can return a solution when complete methods would be too expensive. The complementary question of unsatis ability is generally considered to be harder than that of satis ability (co-NP vs. NP ) [Freuder et al., 1995];1 a systematic method appears to be necessary to rule out the possibility that a portion of the search space contains solutions. Stochastic search is not appropriate, and if we want ecient methods which can answer the unsatis ability question for at least some instances, we must look elsewhere. Abstraction | using single object to represent a whole class of objects | provides one avenue for reducing search complexity while still addressing the unsatis ability question systematically. We investigate abstraction with respect to CSP variable domains (described in Section 2), using a parameterized space of random CSP ensembles as a frame of reference (Section 3) . The abstraction reduces information in a one-sided way, with constraints generally becoming looser, so that the technique only supports determining unsatis ability of the input instance and is incomplete in this regard. The technique is e ective only when the input instance and the reduced instance are both unsatis able. It has been shown previously that the random problem space exhibits a steep gradient in average frequency of unsatis ability as the number of constraints or their tightness is increased while other parameters are held constant [Williams and Hogg, 1994, Smith and Dyer, 1996, Prosser, 1996]. The gradient is a (hyper-)surface of \phase transition" | a sharp boundary between an \under-constrained" region of almostall satis able instances and an \over-constrained" region of almost-all unsatis able instances. The image of a random CSP ensemble under the abstraction can be approximated reasonably as another point in the same parameterized space. Thus, the phase transition surface allows us to characterize when the abstraction will almost always be e ective | when points representing parameters for the input ensemble and the reduced ensemble are both in the over-constrained region. 1
We refer to Bart Selman's position statement in the cited panel summary.
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Qualitatively, the abstraction is e ective when input constraints are very \tight" | that is, they allow relatively few tuples. Quantitatively, we suggest how to predict abstraction e ectiveness using speci c mathematical approximations (Section 4). We show that the loosening of constraints under domain abstraction is accelerated as the degree of abstraction is increased, so that domain abstraction's e ectiveness drops o suddenly. We correct an approximation commonly used to predict the phase transition boundary, to make it perform better in a region with sparse constraint graphs and very tight constraints, for which our abstraction technique is e ective. Then we empirically verify our predictive method in this region of the problem space, which is representative of a class of CSPs arising from relational data base queries. While we perform the bulk of our investigation under a particular abstraction mapping policy, we also discuss how domain abstraction's e ectiveness is a ected by the choice of mapping policy (Section 5). Other abstraction approaches [Freuder, 1991, Ellman, 1993a, Ellman, 1993b, Freuder and Sabin, 1995] are known to be e ective for CSPs which possess the right kind of structure in their variable domains. Our results represent the rst characterization of the e ectiveness of domain abstraction on unstructured, random CSPs. We suggest that domain abstraction with randomly generated mappings like ours can be complementary to the above approaches in a setting of mixed problems where some variable domains have known structure and some have unknown or absent structure.
2 CSP Domain Abstraction Consider a nite-domain CSP [Mackworth, 1992] with n variables X = fx1; x2; : : :; xng, all ranging over a common domain of d values V = fv1; v2; : : :; vdg, and m binary constraints. Each of the constraints allows exactly q tuples. A solution to the CSP assigns a value to every variable such that the corresponding tuple is allowed by every constraint. The worst-case time complexity of any known sound and complete algorithm for solving the CSP in the general case is O(dn ). Domain abstraction transforms an input CSP using a many-to-one mapping to reduce the size of its variable domains by a constant factor ; the new domain size is d0 = bd= c. We require that d0 2 (except in Section 4.6, and we make the simplifying assumption that is an integer which evenly divides d.) Using a sound and complete algorithm to solve the reduced CSP, we see a worst-case complexity of O(d0n ), obtaining a savings factor of O( n ) from domain abstraction. It is clear that greatest eciency gains will come when input domains and reduction factors are large. As our underlying sound and complete algorithm, we choose FC/DVO-FF/CBJ | that is, backtracking search with forward checking; dynamic variable ordering using the fail- rst heuristic, which selects a variable with a smallest remaining domain; 3
and con ict-directed backjumping [Prosser, 1993]. Making this particular choice automatically a ords another combinatorial advantage in solving certain kinds of CSP instances which otherwise we would want be obliged to take steps to exploit. The constraint graph of a CSP instance is formed by treating variables as vertices and binary constraints as edges. In the case of a constraint graph with several disconnected components, some algorithms would perform a multiplicative amount of work, but FC/DVO-FF/CBJ performs no work beyond the sum of the work required to solve each component individually. Our many-to-one abstraction mapping is a function f : V ! V 0. We apply this mapping as follows. For any tuple t = (1; 2), its reduction is f (t) = (f (1); f (2)). For any constraint relation R, its reduction f (R) is the union of the reduction of its tuples. To illustrate, suppose the input domain V = fa,b,c,dg, the reduction factor
= 2, and the reduced domain V 0 = fx,yg. In general, we will construct the mapping for a given CSP instance according to some particular \mapping policy", but here we arbitrarily choose the following mapping f .2 v2V a b c d f (v) 2 V 0 x x y y Figure 1 shows the e ect of this mapping on an individual constraint R(x1; x2) represented in boolean matrix format: a 1 appears at a given index pair when the corresponding constraint tuple is allowed; a 0 otherwise. In making explicit the result of the inverse mapping f ?1 which takes the reduced constraint back to its original, input domain, this example illustrates the e ective loosening of constraints by domain abstraction; this loosening makes domain abstraction sound and incomplete with respect to unsatis ability. This loosening is less likely to result in the transformation of an unsatis able instance into a satis able one when input constraints are tight.
R0 (x1; x2)
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x y 0 1 1 0 a 1 ?1 f f 1 0 0 1 =) x 1 1 =) b 1 0 1 0 0 y 1 0 c 1 0 0 0 0 d 1 Figure 1: The e ect of an abstraction mapping on a constraint. make constraints more permissive (\looser"). a b c d
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In this section we assume that the same mapping f is used for all variables in X . Later when we empirically evaluate our predictions of domain abstraction's e ectiveness (in Section 4) and compare di erent abstraction mapping policies (Section 5), we generate a distinct mapping for each of a CSP instance's variables, according to the policy that is in force. 2
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For any CSP instance I , its reduction f (I ) is the set of the reductions of its constraints. An assignment is a pairing of a variable x and a domain value v: v=x. For any assignment = v=x, its reduction is f () = f (v)=x. An assignment set is a set of assignments which mentions any given variable no more than once. For any assignment set , its reduction is the set of the reductions of its members: f ( ) = ff () : 2 g. For any set of assignment sets , its reduction is the set of the reductions of its members: f () = ff ( ) : 2 g. For a set S , let S j represent the cross product of S with itself j ? 1 times; i.e., S 2 = S S . Represent a tuple of a constraint by its corresponding assignment set. Represent a constraint by its set of allowed tuples. For a reduced tuple t0 2 f (R), let f ?1 (t0) = fu 2 V 2 : f (u) = t0g | the set of possible tuples which map to t0. In general, a reduced assignment set is an assignment set over variables with the reduced domain V 0. For a reduced assignment set 0 = fv1=x1; : : : ; vk=xk g, let f ?1 (0) = fu 2 V k : f (u) = 0g. Theorem: If a reduced instance f (I ) is unsatis able, then the input instance I also is unsatis able. Proof: Consider the combinatorial lattice (see e.g., [Bollobas, 1985]) G formed by all the possible assignment sets for the variables in I , using the partial order of assignment set containment. In this lattice, we de ne the lower shadow L( ) of an assignment set to be the set of all assignment sets which contain : L( ) = f 2 G : g. We de ne the lower shadow of a set Sof assignment sets to be the union of the lower shadows of each member set: L() = 2 L( ). Populate G with the set K of assignment sets corresponding to tuples disallowed by any constraint in I . Let L(K ) be the lower shadow of K in G. For f (I ), the lattice is f (G), and the set of disallowed assignment sets is K 0 f (K ), with lower shadow L(f (K 0)). In e ect, our mapping prefers allowed tuples (not disallowed ones) whenever several possible tuples map to the same value in V 0. Now, if an assignment set 2 L(K 0) then for all assignment sets 2 f ?1( ), 2 L(f (K )). Call the assignment sets corresponding to the disallowed tuples and those in their lower shadows \dark". A CSP is unsatis able i all of the assignment sets of length n are dark. Clearly, if all assignment sets of length n in f (G) are dark (and f (I ) is unsatis able), then also all assignment sets of length n in G are dark (and I is unsatis able). 2
2.1 Related Work Domain abstraction has been suggested, more or less directly, by Imielinski [1987], Sahtaridis [1991], Giunchiglia and Walsh [1992], Dalal and Etherington [1992], and Ellman [1993a, 1993b], but as far as we know it has not previously been evaluated with reference to unstructured problems. Ellman introduced a family of CSP abstraction techniques and showed that these lead to signi cant performance gains in particular 5
cases in which CSPs either possess some explicit hierarchical structure or can be reformulated to re ect a hierarchical structure | as in using a successive re nement of numeric intervals for successive abstractions of a numeric variable domain. Ellman did not evaluate domain abstraction for random, unstructured CSPs, and our results represent the rst characterization of domain abstraction's e ectiveness in this regard. Freuder and Sabin [1995] evaluate domain abstraction with information-preserving mappings based on \interchangeability" among domain values [Freuder, 1991] using random CSP spaces with controlled | non-random, uniformly non-zero | degrees of interchangeability. Other systematic constraint satisfaction approaches which are sound and incomplete with respect to unsatis ability include the method of Schaerf and Cadoli [1995] to examine only a proper subset of an instance's variables, or limited k-consistency processing in hopes of nding an instance k-unsatis able for small k. Although domain abstraction could be combined with the former method, choosing a likely unsatis able subset of variables has been problematic. Recently Mazure et al. [1996] have used failures in GSAT, a non-systematic local search algorithm [Selman et al., 1992], to guide backtracking search toward early detection of unsatis able sub-problems in propositional satis ability; a similar, combined backtracking/local search technique might work for CSPs. With respect to the latter method, if small-k-unsatis ability is expected, the sound and complete algorithm to be used after the abstraction phase can employ static or dynamic small-k-consistency processing. (Forward checking in the FC/DVO-FF/CBJ algorithm enforces 1-consistency dynamically, for example.)
3 Abstraction in a Random Problem Space A random problem space serves as a convenient, controllable framework for evaluating statistical properties of problems and techniques or algorithms pertaining to them. We use a space of random CSP ensembles to characterize the e ectiveness of domain abstraction. Others [Williams and Hogg, 1994, Smith and Dyer, 1996, Prosser, 1996] have investigated the satis ability question in an equivalent space with n slightly di erent parameter de nitions (e.g., Smith's p1 = m= 2 , p2 = 1 ? q=d2).
3.1 Random CSP Ensembles hn; d; m; qi Each random CSP ensemble is speci ed by instantiating fully the 4-tuple hn; d; m; qi with particular integer values, where the parameters are as in the previous section | n variables with domain size d and m constraints each allowing q tuples. Thus, we have a 4-dimensional space of possible ensembles. When convenient, we refer to smaller-dimensional spaces over subsets of the given four parameters, in which we have xed particular values for other parameters. We call a point in the 4-dimensional 6
space an ensemble. To generate a CSP instance for an ensemble, we randomly select m distinct constraints from among the n2 possible binary constraints, according to the uniform distribution; for each of these constraints, we allow q from among the possible d2 tuples, also uniformly. We are interested primarily in the average frequency of unsatis ability exhibited by such an ensemble.
3.2 A Mean Field Approximation Domain abstraction roughly maps one point in the 4-dimensional problem space into another; we employ a mean eld approximation (described below) to treat domain abstraction as if it actually behaved this way. We refer to the problem space that results the application of a particular abstraction mapping policy to the input space as a reduced problem space. The numbers of variables n and constraints m are una ected by the abstraction; the domain size d is reduced by the constant factor to the new domain size d0; and we de ne a random variable Q to represent the number of tuples allowed by a constraint reduced under the abstraction mapping. We approximate the natural probability distribution (\ eld") whose expected value (\mean") is E [Q] by an arti cial distribution whose value is 1 at E [Q] and 0 everywhere else. This replaces the actual reduced ensemble with inhomogenous constraint tightness by a point q0 in our space with xed tightness; that is, q0 = E [Q]. If this approximation is close enough, we have a rst step toward characterizing the e ectiveness of domain abstraction: if almost all the problem instances in both the input ensemble and in the reduced ensemble are unsatis able, then the abstraction technique almost certainly will be e ective in returning the de nite answer. We address the accuracy of this approximation in Section 4.5.
3.3 The Phase Transition Boundary Our random problem space exhibits a steep gradient in average frequency of unsatis ability as the number of allowed tuples q is decreased, or as the number of constraints m is increased, while number of variables n and domain size d are held constant [Williams and Hogg, 1994, Smith and Dyer, 1996, Prosser, 1996]. The gradient is a surface of \phase transition"3 | a sharp boundary between regions of almost-all satis able and almost-all unsatis able instances. This means that we can focus our attention on this boundary (treating it as a step function4) and cover almost all the cases of interest. If the reduced ensemble lies on the unsatis able side of the phase A name given in analogy to the physical phenomenon in which a substance's \phase" (a qualititative description of average inter-molecular distance) changes abruptly, e.g., from a liquid to a gas as its temperature is increased at a constant rate. 4 Empirically it is seen that such phase transitions become steeper with increasing problem size (number of variables) n. 3
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transition | in the over-constrained region | then domain abstraction almost always will be e ective in providing the de nite answer; if the reduced ensemble lies on the satis able side or the transition | in the under-constrained region | then domain abstraction almost never will be e ective. Figure 2 shows the e ect of varying the value of the parameter q, the number of tuples allowed per constraint, while holding other parameters constant in the 1-dimensional space of CSP ensembles h10; 16; 15; qi. The phase transition is evident in the region around q = 47. 1 0.8 0.6
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Figure 2: Part unsatis able for random CSP space h10; 16; 15; qi. As constraint tightness decreases (with increasing values of q), there is an abrupt change | a phase transition | in average frequency of unsatis ability, from near 1 to near 0. Experimental data are from 100 samples/ensemble. Figure 3 includes the curve h10; 16; 15; qi as in Figure 2 and includes additional curves for smaller domain sizes d 2 [4; 15]. We have chosen these curves to illustrate the e ect of domain abstraction when an input ensemble is in h10; 16; 15; qi and the reduced domain size d0 is in the open interval [4; 15]. Remembering that under domain abstraction n and m do not change, we see that varying d and q tells us where domain abstraction will likely be e ective under the mean eld approximation | that is, when both the input and reduced ensembles fall on the unsatis able \plateau" in Figure 3. It will likely be ine ective when the reduced ensemble falls on the satis able \plain" in Figure 3. Figure 4 depicts the phase transition in the 1-dimensional space h10; 16; m; 47i in which m is varied and q is now held constant. The phase transition is evident 8
h10; d; 15; qi
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Figure 3: Part unsatis able for random CSP space h10; d; 15; qi. Each curve corresponds to a di erent domain size d 2 [4; 16]. Under the mean eld approximation q0 = E [Q], the abstraction trajectory (top/left arrow) which stays on the \plateau" is e ective; the trajectory (bottom/right arrow) which reaches the \plain" is ine ective. Experimental data are from 100 samples/ensemble. around q = 15. (This is expected, since the ensemble h10; d; 15; 47i has the same %-satis ability both in this space and in that pro led in Figure 2.) We will make reference to phase transitions along both of these dimensions in our evaluation of domain abstraction's e ectiveness.
4 Predicting Abstraction E ectiveness The phase transition curves presented in the previous section were derived empirically. Now we describe analytical techniques for predicting where domain abstraction with a particular reduction factor will be e ective in the random problem space. 9
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Figure 4: Part unsatis able for random CSP space h10; 16; m; 47i. As number of constraints m increases, there is a phase transition in average frequency of unsatis ability, from near 0 to near 1. Experimental data are from 100 samples/ensemble. Assuming for the moment that the mean eld approximation q0 = E [Q] is reasonable, there are two components necessary to predict the boundary of domain abstraction's e ectiveness: 1. predicting E [Q]; and 2. predicting where an ensemble hn; d0; m; E [Q]i lies with respect to the phase transition surface. We will address each of these predictions in turn (Sections 4.2 and 4.3) and then return to evaluate our predictions empirically (Section 4.4) and analytically (Section 4.5). At this point, we also will be in a good position to discuss predicting the maximum e ective degree of abstraction for a particular ensemble (Section 4.6). First, though, we describe one possible domain abstraction application area which we will use as the context for this evaluation (Section 4.1). We return to discuss the suitability of domain abstraction for this application at the end of this section (Section 4.7).
4.1 CSPs from Data Base Queries Among the prospective applications for domain abstraction are the CSPs that arise as queries in intelligent data base systems. Relational query optimizers [Graefe, 1993] 10
map queries expressed in a query language, e.g., SQL, to a \query graph". This can be viewed as the constraint graph of a CSP: the relevant relational tables (CSP variables) are query graph vertices; the data base tuples that make up the tables are variable domains; and predicates included in the query (constraints) are query graph edges. In distributed data bases, optimizers often apply the relational operator \semi-join" in a pre-processing pass whose result is identical to making variable domains directed arcconsistent [Korth and Silberschatz, 1986, Bayardo Jr., 1995]. This connection between constraint satisfaction and the evaluation of data base queries has been made before (e.g., [Dechter, 1992]) but historically has not been exploited; the number of tables in a typical data base query has been small | on the order of 1 to 4 | and the power a orded by CSP techniques has not been perceived as needed. Current e orts to integrate inference systems with data bases change this situation. The data base literature refers to the integration of forward- and backward-chaining rule systems with data bases as \active" and \deductive" data bases, respectively [Stonebraker, 1992, Ullman, 1988]. In both cases a rule's predicate is represented as a data base query | i.e., a CSP. Such CSPs have very di erent surface characteristics from those typically studied in non-data base contexts. The measurement of expert systems has revealed that the dominating rule predicates map to CSPs with teens of variables [Gupta, 1988]. The integration with data bases further suggests variable domains sizes measured in megabytes or gigabytes | i.e., thousands to millions of values. Consequently, the ultimate feasibility of intelligent data bases is intimately tied to discovering and exploiting techniques for minimizing constraint satisfaction search | such as, possibly, domain abstraction. We choose the 2-dimensional random CSP space h10; 1024; m; qi as representative of such data base query CSPs. We arbitrarily select the reduction factor = 16, giving us a worst-case complexity improvement of 256-fold; this is very close to speed-ups we see in practice on instances with connected constraint graphs using the FC/DVOFF/CBJ constraint satisfaction algorithm.
4.2 Predicting E [Q] The probability distribution for the random variable Q | the number of tuples allowed per constraint in a reduced instance | is dependent on the particular policy followed in generating the mapping functions for abstraction. Below we describe one simple mapping policy, Blind, which is competitive with other mappings we have explored (see Section 5).
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Blind(V; V 0 ) 1 for i 2 [1; d0] begin 2 for j 2 [1; ] begin 3 Add(Choose/Delete(V ), V 0[i]);
4 end; 5 end; Blind maps exactly values (selected randomly with uniform probability) from the input domain V onto exactly one value from the reduced domain V 0, without examining any constraints. We assume that the domains for V and V 0 consist of the integers [1; d] and [1; d0], respectively. V is represented as a linked list and V 0 as an array of linked lists, which initially are empty. V 0 is indexed from 1 to d0. The procedure Choose/Delete randomly selects one element from its list argument, destructively deletes it from the list, and returns it. Add destructively adds its rst argument to its second, list argument. At the conclusion of Blind, each entry of the array V 0 holds a list which represents its pre-image under the randomly constructed mapping f .5 The time complexityof Blind is dependent on that of Choose/Delete. If Choose/Delete always selects the rst member of its list argument, then its complexity is O(1) and that of Blind is O(d0 ) = O(d). We can do this and still achieve randomness by rst randomly permuting the list V , in O(d log d) time. Calculating the exact probability distribution for Q under Blind appears to be a complex task. However, the binomial distribution in which each of d02 tuples is 2 allowed with probability p = 1 ? (1 ? q=d2) is a good approximation. (Each possible tuple in the input constraint is allowed with probability q=d2; each possible reduced tuple is the \target" of 2 of the possible input tuples.) The binomial approximation introduces its error which has the greatest a ect on the mean value E [Q] by according non-zero probabilities to values of Q which cannot appear, such as to the value d0 when q < d0, or to values less than d0 when q > d2 ? 2. The error in E [Q] for such extreme values of q, however, always amounts to less than 1 allowed tuple in the reduced constraints. For more moderate values of q, we have found that the mean derived from the binomial approximation di ers negligibly from natural means determined empirically. In the following, we approximate the2 natural mean as the mean of the binomial distribution, E [Q] = d02(1 ? (1 ? q=d2) ). Figure 5 shows how quickly a reduced constraint is expected to become a \nonconstraint" under Blind. Here we have d = 1024 and = 16 (so, d0 = 64), with q varying. Allowing merely 1/32 (that is, 16,384) of the possible d2 = 1; 048; 576 input tuples leads to an expectation that (nearly) all of the d02 = 4096 reduced tuples will be allowed | practically neutralizing all constraints. Under the mean For the presented cases in which evenly divides d, Blind is equivalent to the \Random Clustering" mapping policy used by Ellman [1993b]. 5
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eld approximation q0 = E [Q], this policy results in the mapping of an input instance 0 from the over-constrained region into the phase transition region when E [Q] = q50% | the number of allowed tuples at the 50%-satis able point in the reduced problem 0 also depends on the space which is 1-dimensional in q. (The speci c value for q50% values used for n and m in the reduced problem space.) When the attempted degree 0 , the input of abstraction and resulting loosening are still greater, so that E [Q] > q50% instance maps beyond the phase transition region into the under-constrained region; the reduced instance is almost certainly satis able, so that domain abstraction is almost certainly ine ective. 4096 1024 256
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Figure 5: Expected reduced tuples allowed for input tuples q, given input domain size d = 1024, reduction factor = 16, so reduced domain size d0 = 64, under mapping policy Blind. We employ the binomial approximation for E [Q]. Figure 6 shows how the rate of increase in the looseness of reduced constraints varies with for the smaller domain size d = 16. Except for the degenerate case of = 1, each constant- curve has a \plateau" | where E [Q] approaches d02 | over which constraints are practically neutralized. When constraints are neutralized, instances are almost certainly satis able, and domain abstraction is almost certainly ine ective. The existence of these plateaus, which are reached earlier in the q dimension as the reduction factor is increased, is the main reason why domain abstraction in signi cant degrees is e ective only for very tight constraints. For any non-degenerate 0 will occur at some value of E [Q] which falls before this plateau reduction factor, q50% on the \rising slope" segment whose length shrinks with increasing . 13
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Figure 6: Expected reduced tuples allowed for input tuples q, given input domain size d = 16 for various reduction factors , under mapping policy Blind. We employ the binomial approximation for E [Q].
4.3 Predicting the Phase Transition Surface Predicting the exact location and shape of the phase transition surface is beyond current analytical techniques. Let M be a random variable representing the number of solutions (\models") of an instance in an ensemble; the expected number of solutions E [M] = dn (q=d2)m. There are dn possible solutions; each constraint independently allows q=d2 of these. Williams and Hogg [1994] have developed an approximation | which we will refer to as the \standard" approximation | that 50%-satis ability occurs for an ensemble when E [M] = 1: (1) Smith and Dyer [1996] provide bounds on %-satis ability based on E [M] and 2[M] | the variance in number of solutions. From these bounds, it appears that the standard approximation can be quite good provided domain sizes are large enough and constraints are not too tight. While these bounds are too wide to be useful for very small domain sizes, empirical results suggest that the standard approximation may remain fairly good (for fairly loose constraints) except for the smallest non-trivial domain size d = 2: for 3- and 4-colorability (i.e., random CSP spaces with d = 3 and d = 4, respectively), the standard approximation calculates m50% within about 0% and 7% of the observed value; for propositional satis ability (i.e., a random CSP space with d = 2), it calculates m50% only to within about 22% [Williams and Hogg, 1994]. So, while this approximation can be very accurate, it also appears to be the 14
least uniformly reliable of our predictive components. Our prospective data base query application has very large domains, but we have seen already that domain abstraction will be e ective only when constraints are very tight | at least tight enough to disallow 31/32 of all possible tuples for d = 1024 and
= 16. For such a tightly constrained input problem space (e.g., h10; 1024; m; 32i) 50%-satis ability tends to occur (in the m dimension) at an ensemble with so few constraints that some variables will be unconstrained completely, and some constraints will mention only variables which are mentioned by no other constraints. In either case, these variables and constraints do not a ect the unsatis ability of a problem instance;6 we refer to them as \isolated". Put precisely, an isolated vertex is a vertex upon which no edge is incident; an isolated edge is an edge whose removal would result in a pair of isolated vertices (it is the only edge incident upon either vertex). Let the random variable V represent the number of isolated vertices in the constraint graph and E the number of isolated edges. Since isolated vertices and edges do not a ect unsatis ability, we expect them to contribute a factor of dE[V ] qE [E ] solutions to those a orded by the remainder of the constraint graph. We o er the \corrected" approximation that 50%-satis ability occurs when E [M] = 1: (2) E [ V d ] qE [E ]
The expected number of isolated vertices E [V ] = n(1 ? m= n2 )n?1 : there are n ? 1 possible edges that could be incident upon a given vertex, each chosen with probability m= n2 . The expected number of isolated edges E [E ] = m(1 ? m= n2 )2(n?2): there are n ? 2 possible edges which could be incident upon either of the 2 variables of a given constraint. Figure 7 plots the expected values of these random variables for n = 50. As a case in point, consider the 1-dimensional problem space h10; 16; 15; qi which is the subject of Figure 2. We are interested in predicting q50% | the number of allowed tuples occurring at the 50%-satis able point in a 1-dimensional problem space where q varies. Solving Equation 2 for q, we predict
E [E ]
!
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; q50% = dn?2m?E[V ] which yields the value q50% = 47:90. Solving instead the uncorrected Equation 1, we would predict q50% = d2?n=m , which for h10; 16; 15; qi yields q50% = 40:32. Empirically (over 1000 instances/ensemble), we nd h10; 16; 15; 47i to have a %-satis ability of 48.4 and h10; 16; 15; 48i to have a %-satis ability of 55.2. Thus, the corrected approximation predicts a value which is much closer to the observed value than does the standard approximation. 6
We require that constraints allow at least 1 tuple.
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Figure 7: Expected numbers of isolated vertices (random variable V ) and edges (random variable E ) for constraint graphs in our problem space which have a total number of vertices (CSP variables) n = 50. In another case illustrating prediction in a di erent dimension, consider the problem space h10; 16; m; 47i which is the subject of Figure 4. We are interested in predicting m50% | the number of constraints occurring at the 50%-satis able point in a 1-dimensional problem space where m varies. Now there appears to be no closed form solution based on Equation 2, but an iterative numerical approach based on it yields the value m50% = 16:01. Solving the uncorrected Equation 1, we would predict m50% = logq=d2 d?n , which for h10; 16; m; 47i yields m50% = 16:36. Empirically (over 1000 instances/ensemble), we found h10; 16; 14; 47i to have a %-satis ability of 72.5 and h10; 16; 16; 47i to have a %-satis ability of 22.0. We found h10; 16; 15; 47i to have a %-satis ability of 48.4, as we noted above. So again, the corrected approximation predicts a value which is closer to the observed value than does the standard approximation. For sparser constraint graphs, the correction is even more important. The ensembles of constraint graphs generated in our problem space for given numbers of vertices/variables n and edges/constraints m follow Palmer's Model B for random graphs [1985], and we may apply corresponding results from the theory of random graphs. It is known that there is a phase transition for connectedness whose location as n ! 1 is given by the threshold function m = (n=2) log n: almost all graphs with more edges are connected; almost all graphs with fewer are not. Figure 8 plots this threshold function for some nite values of n. We see that its value of m = 97 for n = 50 correlates fairly well with the disappearance of isolated vertices in Figure 7. The theory of random graphs helps to explain this correlation: there is also 16
a threshold function at m = n=2 for the existence of a unique \giant component", which includes signi cantly more vertices than any other component. As m increases between these two thresholds, the largest components become absorbed into the giant soonest, so that for graphs just at the edge of connectedness, almost all of their remaining components are isolated vertices.7 Thus, while isolated vertices have their greatest prevalence in the sparsest constraint graphs, they also are important for graphs that are very nearly connected, and by correcting for them alone we may safely apply techniques appropriate for connected graphs, as we have done for phase transition location estimation. 4096
Connectedness Giant Component
1024 256
m
64 16 4 1 0.25
2
4
8
16
32
n
64
128
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Figure 8: Asympotic thresholds for constraint graph connectedness and for the existence of a unique giant component in random graphs from Palmer's Model B, plotted for some nite values of n.
4.4 Combining and Testing the Predictions When we put the two component predictions together, we obtain the overall prediction that 50%-satis ability occurs in the reduced space when E [M0] (3) d0E[V ] E [Q] E [E ] = 1; where E [M0] = d0n (E [Q]=d02)m . Figure 9 plots predicted regions of e ectiveness for domain reduction against actual experimental results. The prediction using the Isolated edges, the next-simplest components (of size 2), disappear in this progression just before the isolated vertices do. 7
17
corrected approximation follows the observed curve quite closely. Domain abstraction will be e ective for a given number of allowed tuples q when the actual number of constraints m is greater than the corresponding plotted value of m050% | the number of constraints at the 50%-satis able point in the reduced problem space which is 1dimensional in m. For ensembles with numbers of allowed tuples q 2048, we found that m050% did not occur, because more than 50% of the ensembles' instances are n satis able given the maximum number of constraints m = 2 = 45. We also note that it was infeasible to solve 100 of the input, unreduced instances of the ensembles with the largest numbers of allowed tuples, so domain abstraction really is putting some dicult unsatis ability questions within reach.
h10; 1024; m; qi; = 16
30 25 20
m0
50%
+ 3
15 10 5+ 0
+ 3 2
Corrected 3 Standard + Observed 2
+
+
+
3 2
3 2
2 3
3 2
4
8
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+
+ 3
64
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2
3 2 q
+ 3
2
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1024
Figure 9: Observed and predicted regions of e ectiveness for domain abstraction under mapping policy Blind; predictions using both the standard and corrected approximations are given. Experimental data are from 100 samples/ensemble.
4.5 Evaluating the Approximations To recapitulate, in this section we have employed three main approximations. The approximation embodied in Equation 1 for locating the satis ability phase transition surface has been used by others [Williams and Hogg, 1994, Smith and Dyer, 1996, Prosser, 1996]; we have improved this approximation with the corrected Equation 2. We have introduced two new approximations speci cally to predict the e ect of abstraction on the phase transition surface; these are mean eld approximation q0 = E [Q] and the binomial approximation for the distribution of the random variable Q. We brie y evaluate the latter two approximations in this section. 18
We can assess the quality of the mean eld approximation by comparing the satis ability phase transition which occurs naturally in a reduced problem space to the phase transition which occurs for an unreduced space which under the mean eld approximation represents that reduced space. (We will do so in a moment.) We can assess the approximating quality of the binomial distribution by examining the phase transition for a random problem space where this distribution occurs naturally. To do so, we de ne the problem space hn; d; m; qiB : instances are formed as for hn; d; m; qi, except that instead of each constraint's allowing exactly q tuples, each of a constraint's possible tuples is allowed with probability q=d2. (If we happen to get a constraint which allows 0 tuples, we throw that whole instance out, so that no instances are trivially unsatis able.) Figure 10 depicts the phase transitions for h10; 16; 15; qi, for h10; 16; 15; qiB , and for several reduced problem spaces with larger input domain sizes and reduction factors. All of the depicted problem spaces are represented under the mean eld approximation by the same problem space h10; 16; 15; E [Q]i. Both of the approximating problem spaces are shown paired with the degenerate reduction factor
= 1, so that we can compare all of the depicted spaces using the common measure E [Q] on the horizontal axis. From Figure 10, it appears that the mean eld approxi0 for the reduced spaces and mation slightly under-estimates part unsatis able and q50% that the binomial distribution may very slightly over-estimate part unsatis able and 0 . Either approximation appears to be reasonable; the binomial approximation q50% seems better.
19
1 0.8
+ 34 2+
3
+
0.6
Part Unsat.
0.4 0.2 0
h10; 16; 15; qi; = 1 3 h10; 16; 15; qiB ; = 1 + h10; 64; 15; qi; = 4 2 h10; 256; 15; qi; = 16 h10; 1024; 15; qi; = 64 4
34 24+ 24
40
45
24 3 +24 3 2+4 + 3 2 2 4 4+ 3 24 + 3 324 2+4 3 24+ 3 50
E [Q]
55
60
Figure 10: Empirical rationale for the mean eld approximation q0 = E [Q] and for the binomial approximation for the distribution of the random variable Q. We plot the satis ability phase transitions for the problem space h10; 16; 15; qi corresponding to a mean eld approximation, for the problem space h10; 16; 15; qiB corresponding to the binomial distribution | both labeled here with the degenerate reduction factor
= 1. We also plot the transitions for a few reduced problem spaces which result from larger input domains and reduction factors. On the horizontal axis we use the 0 show binomial approximation for E [Q]; empirical checks of the natural means at q50% negligible di erences. (Note that the rst (bottom) curve is the same problem space as pro led in Figure 2, except at a di erent scale.) Experimental data are from 1000 samples/ensemble, except for the reduced problem space h10; 1024; 15; qi under
= 64, where we used 100 samples/ensemble. The quality of the binomial approximation indicated by Figure 10 suggests that we could rely on this approximation alone | foregoing the mean eld approximation | if we could derive the expected number of solutions E [MB] for an ensemble hn; d; m; qiB . Then we could substitute this value for E [M0] in Equation 3. For an ensemble hn; d; m; qi, the expression E [M] = dn (q=d2)m is exact, but for an ensemble hn; d; m; qiB , it is not. Let the random variable QB represent the number of tuples allowed by a constraint of an instance in hn; d; m; qiB . In the expression for E [MB ], we should include a separate factor of qi=d2 for each reduced constraint with number of allowed tuples QB = qi. In the binomial probability distribution, each of d2 tuples is allowed with probability p = q=d2. Thus, the probability that QB takes
20
on a particular value qi for qi 2 [1; d2] is8
!
2 P (QB = qi) = dq pqi (1 ? p)d2 ?qi : i Noting that each term qi=d2 is to appear in a product, we need a kind of expectation based on the geometric mean rather than oneP based on the arithmetic mean, as 2 d E [ln Q ] B follows: EG[QB ] = e , where E [ln QB ] = qi=1 P (QB = qi) ln qi. Then we may use the exact expression E [MB ] = dn (EG [QB]=d2)m. We have chosen not to use the exact expression for E [MB ] here because the mean eld approximation appears to be adequate for our examples, because the exact expression precludes deriving a closed-form expression for q50% or for m50%, and because calculating values for E [MB] is numerically challenging when d is large.
4.6 Predicting Maximum E ective Degree of Abstraction So far in this section, we have used Equation 2 to predict where the 50%-satis able point will be under a given degree of abstraction, or reduction factor, . Now, we turn this question around and ask, \For a given ensemble, or point in the random problem space, what degree of abstraction will lead to 50%-satis ability in the reduced space?". When we treat the phase transition as a step function, this amounts to asking, \What is the maximum reduction factor we can use and still expect domain abstraction to be e ective?". Of course, the input ensemble must be mostly unsatis able for the question to make sense. Figure 11 shows the phase transition that occurs for the ensemble h10; 1024; 15; 32i under mapping policy Blind as the reduction factor is varied over its allowable range of [1; 512]. In e ect (and only in this section), we are considering the reduction factor as a fth parameter in the (now 5-dimensional) problem space. Here we plot
using a log scale because as a reduction factor it has a multiplicative e ect. Figure 12 shows predicted and observed values of 50% | the reduction factor at which 50%-satis ability occurs in the 1-dimensional problem space where varies. Predictions are based on Equation 2; again there appears to be no closed form for 50%, and we use an iterative numerical method. Observed %-satis able for our predicted
50% values are listed in the following table.9 Ensembles with the largest numbers of tuples q | 1024, 2048, and 4096 | were too dicult for our experimental method The dummyvariable i ranges over [1; d2] rather than [0; d2] because we require that all constraints allow at least one tuple. If we relax this requirement, we must deal with the case of qi = 0 separately. When qi = 0, we know that the resulting instance will have zero solutions. Then we reduce our expected number of models E [MB ] (as de ned below) by a factor of (1 ? P (QB = 0))m | the probability that an instance does not have zero models because some constraint in it allows zero tuples. 9 In the table, experimental data are from 100 samples/ensemble. 8
21
13
h10; 1024; 15; 32i under mapping policy Blind
3
3
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1
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Figure 11: Part unsatis able for random CSP space h10; 1024; 15; 32i. As the reduction factor increases, there is a phase transition in average frequency of unsatis ability, from near 1 to near 0. Experimental data are from 100 samples/ensemble. to handle with the sought-for, low reduction factors, and we were unable to gather data for them. q 4 8 16 32 64 128 256 512 %-sat. 0 0 15 26 22 13 21 29 Observed and predicted values in Figure 12 agree fairly well; the largest deviation (for q = 8) occurs when the reduced problem has the smallest possible domain size d0 = 2 at observed 50% . As we see from the table above, all of the other predicted 50% values at least fall into the general region of the phase transition for . Below, we discuss reasons for this deviation at smallest d0 in the context of the approximation evaluations given in the preceding section. When d0 = 2, which corresponds to the CSP special case propositional satis ability, we know that Equations 1 and 2 are weak as predictors of 50%-satis ability [Williams and Hogg, 1994]; 50%-satis able ensembles tend to have signi cantly more than one solution [Selman et al., 1996]. For the case of q = 8, we have E [Q] = 3:459, and for the problem space h10; 2; 15; 3:459iB | with binomially distributed values for q | we observe a %-satis able near that we do for h10; 1024; 15; 8i with = 512 under Blind (9 vs. 20);10 this suggests that the binomial approximation is not at fault for the deviation. The exact expected number of solutions E [MB ] for the binomial problem space is slightly lower than the expected number of models under the mean 10
Once again, experimental data are from 100 instances/ensemble.
22
512 + 256 128
50%
h10; 1024; 15; 32i under mapping policy Blind
3 +
3 +
64 32 16 8 4 2
4
16
Observed 3 Predicted +
3 +
3 +
64
3 +
q
3 +
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+ 1024
+
+
4096
Figure 12: Observed and predicted maximum reduction factors for random CSP space
h10; 1024; 15; qi. The observed data are from experiments using 50 samples/ensemble in a binary search pattern until a %-satis ability in the open interval (45,55) was observed.
eld approximation E [M] (80 vs. 116), and the left-hand side of Equation 2 comes closer to the predicted value of 1 (65 vs. 94), but still not very close. As we suggested earlier, the phase transition location prediction seems to be the weakest link in the chain of approximations.
4.7 Evaluating Domain Abstraction for Data Base Queries The results of the preceding section show that dramatic complexity reductions are possible when constraints are very tight. Domain abstraction acheives a complexity savings of O( n ) when constraint graphs are connected. We note that instances explored in the previous section are fairly highly connected: with 10 vertices in the constraint graphs, the expected number of isolated vertices E [V ] = 0:26. As we have already mentioned, even modest complexity reductions seem to require a tightness that might be considered extreme. At this point, it is impossible to predict how tight the constraints will be that arise in query CSPs for intelligent data bases.11 The distriWe are being rather casual with the term \tightness" here. Others (e.g., [Smith and Dyer, 1996]) de ne tightness formally as 1 ? q=d2. The fact that domain abstraction is e ective for a given (formal) tightness in one problem space does not mean that it also will be e ective for the same tightness in another problem space; in general, separate analyses must be performed to determine the e ectiveness range for each problem space considered. 11
23
bution of such queries almost certainly will not match any given point in our random problem space. Queries from real-world applications tend to include more structure than random queries do, so perhaps the interchangeability approach of Freuder and Sabin [1995] which can identify exact symmetries which arise serendipitously can be helpful. Also, real-world data bases usually (but not always) have data dictionaries which include much information about types in relational tuples (corresponding to CSP domain values), so perhaps the approximate symmetry approach of Ellman [1993a, 1993b] also will be useful. When conditions are favorable, domain abstraction with randomly generated mappings like ours might be appropriate. In summary, we think it is worth exploring all three kinds of approaches to reducing complexity for the CSPs arising from data base queries.
5 Alternative Abstraction Policies Given that the phase transition surface is an inherent, unchanging property of the random problem space hn; d; m; qi, the only opportunity we have to improve upon domain abstraction's e ectiveness lies in developing alternative abstraction mapping policies. In this section, we describe two alternative policies which we have explored. Both of these policies are more expensive than Blind, taking at least O(d log d) time. We describe rst and in more detail a weight-based policy, Even, whose e ectiveness was slightly better than that of Blind. Then we describe a bottom-up clustering policy, Clustered. We compare the performance of each of the three policies in Figure 13 on the same task as is evaluated in Figure 9. Once again (as in Figure 9), domain abstraction will be e ective for a given number of allowed tuples q when the actual number of constraints m is greater than the corresponding plotted value of m050%. Under the policy Even, for ensembles with numbers of allowed tuples q 4096, we found that m050% did not occur, because more than 50% of the ensembles' instances are satis able given the n maximum number of constraints m = 2 = 45. Under the policy Clustered, this condition occurs much earlier | for q 256. Together, the results for these two policies suggest that there may be little opportunity to improve upon the performance of Blind for random problems. We suspect that the saturation of reduced constraints for Blind occurring on the constant- plateaus of Figure 6 may be a ubiquitous characteristic possessed by any mapping policy. In instances with very few allowed tuples (very small q), exploiting the opportunities a orded by local information can forestall this saturation to some degree, but as more tuples are allowed the saturation may be inevitable. It remains an open question whether policies like these might be e ective for more structured problems (such as the CSPs from data base queries investigated in Section 4). 24
h10; 1024; m; qi; = 16
45 40 35 30 Blind 3 25 Even + 0 m50% 2 Clustered 20 15 10 5 3 3 + 3 2 + + + 3 2 2 2 0 4 8 16 32
+
3 +
2 2 3
3 +
+
64
q
128
3 + 256
3 +
512
1024 2048
Figure 13: Observed regions of e ectiveness for domain abstraction under three different mapping policies. Experimental data are from 100 samples/ensemble.
5.1 A Value Weight-based Policy The policy Even attempts to ensure that the pre-image of each value in V 0 represents approximately the same total number of occurrences of values in V .
25
Even(V; V 0 ) P 1 quota ( fv : (v; w) 2 V g)=d0; 2 i0 1; 3 while (Not-empty(V ) ^ (i0 d0)) begin 4 pair Pop(V ); 5 allocation Weight(pair); 6 Add(Value(pair), V 0 [i0]); 7 while (Not-empty(V ) ^ (Weight(First(V )) + allocation) < quota) begin 8 pair Pop(V ); 9 allocation allocation + Weight(pair); 10 Add(Value(pair), V 0[i0 ]);
11 end; 12 i0 i0 + 1; 13 end; 14 while (Not-empty(V )) begin 15 for i0 2 [1; d0] begin 16 Add(Pop(V ),V 0 [i0]); 17 end; 18 end; Upon entry, V is an list of pairs (v; w), ordered by increasing w, where v is a particular input value and w is the total number of allowed tuples in which v appears over the instance's m constraints (w is the \weight" of v). The procedure Not-empty returns \true" if its list argument is not empty; First returns the rst element of its list argument; Pop destructively removes and returns the rst element of its list argument; and Weight and Value return the rst and second members of their pair argument, respectively. Other procedures and conventions are as for Blind. The policy Even xes a common quota for the combined weights of each pre-image and allocates values from V to a pre-image, until doing so would exceed this quota; it then moves on to construct the pre-image for the next value in V 0. Any excess values are assigned amongst the entire set of pre-images in a round-robin fashion (Lines 14 to 18). The policy Even groups input values into pre-images which are similar to each other in total weight. The values in each pre-image are not biased to have any similarity to each other, however. Ellman [1993b] has shown for structured CSPs that abstractions which map similar input values together in the pre-image of a given reduced value can be very e ective. In our unstructured problem space, we have very limited information about input values. One attribute of values whose similarity we can exploit is their relative frequencies of occurrence; we used this as the basis of a bottom-up clustering abstraction mapping policy, described below.
26
5.2 Bottom-up Clustering Policy The policy Clustered is intended to avoid removing sources of unsatis ability from the problem instance: if certain values for a certain variable are disallowed by a certain constraint, we try not to destroy this information by mapping these values together with other, allowed values for the same variable. Unfortunately, this is not always possible, since di erent values may be allowed by di erent constraints, so that local information alone does not suce. We approximate based on global information: for each value of each variable which actually is mentioned in an instance, we record the ratio of how many times it is allowed by constraints to the number of times (md) it possibly could have been allowed | this serves as a preliminary indication of the constraining power of each value.12 Constraining power falls o exponentially with increasing frequency of a value's occurrence, so we take the logs of the ratios we have collected as our normalized measure of constraining power. Then we cluster the values for a variable by constraining power as described below. For each variable, we want to cluster together values that have approximately the same constraining power, to form d0 separate groups. This is an optimization problem: we choose as our objective function to minimize the standard deviation of the constraining powers in each cluster. We use an irrevocable, bottom-up control policy: start with each normalized value in its own separate cluster, and in successive iterations merge those two clusters which give the best score relative to the objective function, until we have exactly d0 clusters left.
6 Conclusion At rst glance, one might assume that abstractions would degrade gracefully, becoming less useful only gradually as problems move further from the abstractions' regions of primary e ectiveness. We have shown for domain abstraction in a random problem space that the degradation occurs very suddenly. The combined e ects of information loss and a phase transition with respect to satis ability lead to a sharp threshold for abstraction's e ectiveness. Our overall evaluation of domain abstraction for these random problems is mixed: one the one hand, we see that dramatic reductions in the complexity of constraint satisfcation do occur; on the other hand, complexity reductions of any amount come only when input constraint are very tight. Our mixed results for domain abstraction in the random problem space are in contrast to positive results for CSPs whose variable domains have an identi able hierarchical structure [Ellman, 1993a, Ellman, 1993b] or exhibit some degree of interchangeability [Freuder, 1991, Freuder and Sabin, 1995]. Our results serve primarily as an analytical If a value is never allowed, we may act as if it never existed: the e ective domain size for this variable is smaller than d. 12
27
and empirical characterization of the e ectiveness of domain abstraction for unstructured problems. We believe domain abstraction with randomly generated mappings like ours can be complementary to the above approaches in a setting of mixed problems where some variable domains have known structure and some have unknown or absent structure. At the same time, our results do seem to suggest that ignoring information about problem structure when it is available may be a poor short-cut to take in designing a CSP abstraction method.
Acknowledgements We thank Roberto Bayardo for his implementation of the FC/DVO-FF/CBJ constraint satisfaction algorithm and for help with using it. We thank Jimi Crawford, Michael Dent, Rob Holte, Greg Provan, and Ron Rymon for their comments on earlier versions of this paper. We thank the organizers and participants at SARA-95 for their feedback and in uence on this work. We thank V. B. Balayoghan for his helpful answers to questions about mathematics.
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