Approximation Algorithms for Temporal Reasoning Peter van Beek ...

Approximation Algorithms for

Temporal Reasoning

Peter v a n Beek Department of Computer Science University of Waterloo Waterloo, Ontario C A N A D A N2L 3G1

Abstract We consider a representation for t e m p o r a l relations between intervals i n t r o d u c e d by James A l l e n , and its associated c o m p u t a t i o n a l or reasoning p r o b l e m : given possibly indefinite knowledge of the relations between some i n t e r v a l s , how do we c o m p u t e the strongest possible assertions about the relations between some or all intervals. D e t e r m i n i n g exact solutions to this p r o b l e m has been shown to be (almost assuredly) i n t r a c t a b l e . A l l e n gives an a p p r o x i m a t i o n a l g o r i t h m based on c o n s t r a i n t p r o p a g a t i o n . We give new approximation algorithms, examine their effectiveness, and determine under w h a t conditions the a l g o r i t h m s are exact.

1.

Introduction

A l l e n (1983) gives an algebra for representing and reasoning a b o u t t e m p o r a l relations between events represented as i n t e r v a l s . Possible a p p l i c a t i o n areas of the algebra include n a t u r a l language processing ( A l l e n , 1984; Song a n d C o h e n , 1988), p l a n n i n g (Allen and K o o m e n , 1983), and a p a r t of a knowledge representat i o n language ( K o u b a r a k i s et al., 1989). T h i s algebra has been cited by others for its s i m p l i c i t y and ease of i m p l e m e n t a t i o n w i t h c o n s t r a i n t p r o p a g a t i o n algor i t h m s . T h e elements of the algebra are sets of the seven basic relations t h a t can hold between t w o intervals, a n d t h e i r converses.

Th ere is a n a t u r a l graphical n o t a t i o n where the vertices represent intervals and the directed edges are labeled w i t h elements f r o m the algebra representing the set of possible relations between the t w o intervals. Here is an example.

W h e n the relationship between t w o intervals is ambiguous or indefinite we label the edge w i t h the set of all the possible relations. So in our example, i n t e r v a l A either overlaps or starts i n t e r v a l B ( b u t n o t b o t h since the t h i r t e e n basic relations are m u t u a l l y exclusive). L e t {I} be the set of all basic relations, { b , b i , m, m i , o, o i , d, d i , s. si, f, f i , eq}. T h e set of all possible labels on edges is 2 { I } , the power set of { I } . A n y edge for w h i c h we have no direct knowledge of the relationship is labeled w i t h { I } ; hence, the graphs are complete. Inference is done in this scheme t h r o u g h composition of relations: given a relation between A and B and between B and C we can compute a constraint on the relation between A and C. D o i n g this for our example we determine t h a t our knowledge of the relationship between A and C can be strengthened to { b } . To see t h a t this is t r u e we show the t w o possible arrangements of the intervals along an i m a g i n a r y time line.

A overlaps B in the d i a g r a m on the left, A starts B in the one on the r i g h t , and B meets C in b o t h . We see t h a t in b o t h diagrams A is before C. Hence the result.

1.1.

Statement of the Problem

Suppose we are given a set of events, represented as the intervals they occur over, and knowledge of the relationships between some of the intervals. T h e problem is to make explicit the strongest possible assertions about the relationships between intervals. We now make this somewhat more f o r m a l . G i v e n is a directed graph w i t h labels on the edges f r o m the set of elements of the i n t e r v a l algebra. A consistent singleton labeling of the graph is a labeling where it is possible

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to map the intervals to a time line and have the single relations between intervals hold (as in the example above). The minimal label corresponding to a label consists of only the elements of t h a t label capable of being p a r t of a consistent singleton labeling of the graph. The problem t h e n is to determine the minimal labels, removing only those elements f r o m the labels t h a t could not be p a r t of a consistent singleton labeling. Call this the minimal labeling problem (MLP). V i l a i n and K a u t z (1986) show t h a t determining an exact solution to the M L P is N P - h a r d . This strongly suggests t h a t no polynomial time a l g o r i t h m exists. Supposing t h a t we still wish to solve instances of the problem, several alternatives present themselves: • Exponential algorithms: Solve the problem exactly b u t devise efficient exponential algorithms. These may still be practical even t h o u g h their worst case is exponential. Valdes-Perez (1987) gives a dependency-directed b a c k t r a c k a l g o r i t h m , b u t it only finds one consistent singleton labeling of the graph or reports unsatisfiability. • E a s y s p e c i a l cases: Interesting special cases of an N P - H a r d p r o b l e m may be solvable in polynomial t i m e . This alternative often takes the f o r m of l i m i t ing the expressive power of the representation language. • A p p r o x i m a t i o n a l g o r i t h m s : Solve the problem approximately using an a l g o r i t h m t h a t is guaranteed polynomial. T h a t is, design algorithms t h a t do not behave b a d l y — i n terms of the quality of the produced solution—too o f t e n , assuming some probabilistic d i s t r i b u t i o n of the instances of the problem. Allen's (1983) 0 ( n 3 ) a l g o r i t h m is j u s t such an approximation a l g o r i t h m . 1.2.

Overview

In this paper we explore the latter t w o alternatives: algorithms for c o m p u t i n g approximations to the minimal labels between intervals and some special cases where a p p r o x i m a t i o n algorithms are exact. We consider t w o versions of the problem: an all-to-all version where we compute the m i n i m a l labels between every pair of intervals, and a one-to-all version where we determine the minimal labels between one i n t e r v a l and every other i n t e r v a l . A l l e n gives an approximation algorithm for the all-to-all problem based on constraint propagation. We present an a l g o r i t h m for computing better approximations for the all-to-all version of the problem and an a l g o r i t h m for the one-to-all version of the problem. For both versions of the problem we identify easy (polynomial time) special cases where the algorithms are exact, give the results of some computational experiments, and propose a test for predicting when the approximation algorithms are useful. For the all-to-all problem, identifying easy cases involves first giving a counter example to a result in the l i t e r a t u r e . An extended version of this paper proofs is available (van Beek, 1989).

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t h a t includes

2. The All-to-All Problem The minimal labeling problem ( M L P ) is related to the constraint satisfaction problem (CSP) ( M o n t a n a r i , 1974, M a c k w o r t h , 1977). Tsang (1987) and L a d k i n (1988) discuss how an M L P can be viewed as a CSP. The algorithms and results developed for the CSP can then also be applied to the M L P . M a c k w o r t h (1977) discusses approximation algorithms for the CSP, called consistency algorithms, t h a t remove local inconsistencies t h a t could never be p a r t of a global solution. One, t w o , and three consistency are generally referred to as node, arc, and p a t h consistency, respectively. Freuder (1978, 1982) generalizes this to k-consistency and defines strong k-consistency as j-consistent for all j < k. It can be shown t h a t for the interval algebra, strong Ar-consistency is equivalent to ensuring t h a t , for every choice of k of the n vertices, every element of the associated labels is capable of being p a r t of a consistent singleton labeling of the subgraph of k vertices. Strong n-consistency then ensures the labeling is minimal. 2.1. T h e P a t h Consistency A l g o r i t h m Allen's algorithm is a special case of the path consistency algorithm for constraint satisfaction. Descriptions of the p a t h consistency a l g o r i t h m can be found in (Allen, 1983 and M a c k w o r t h , 1977). To use the p a t h consistency a l g o r i t h m we need to define the operations of intersection and composition of t w o labels. Intersection is j u s t set intersection. Let C ij be the label on the edge between i n t e r v a l i and interval j. Given t h a t labels on edges can represent a disjunction of possible relations between t w o intervals, Allen defines the composition of t w o labels as the pair-wise multiplication of the elements, Cik . C k j { a X b I a € Cik, b € Ckj )

(2.1)

where X is defined over the seven basic relations and their converses and is easily implemented as a table lookup (see A l l e n , 1983 for the complete table). 2.2. I m p r o v i n g t h e A p p r o x i m a t i o n In general, Allen's a l g o r i t h m , being an approximation a l g o r i t h m , w i l l not always compute the m i n i m a l label between t w o intervals. In this section we explore better (and, u n f o r t u n a t e l y , more expensive) approximation algorithms. We develop an 0 ( n 4 ) consistency a l g o r i t h m . T h e labels computed by the a l g o r i t h m , as w i t h Allen's a l g o r i t h m , w i l l always be a superset (not necessarily proper) of the m i n i m a l labels. B u t the a l g o r i t h m computes a better approximation in t h a t fewer disjuncts remain t h a t could not be p a r t of a consistent singleton labeling of the g r a p h . Recall the definition of a m i n i m a l label: every element of the label is capable of being p a r t of a singlet o n labeling of the entire graph t h a t can be consistently mapped to a time line. It can be shown t h a t for the i n t e r v a l algebra the p a t h consistency algor i t h m , as an a p p r o x i m a t i o n , ensures the labels are m i n i m a l w i t h respect to all subgraphs of size three.

Inputs A matrix C where Cij is the label on edge ( i , j). O u t p u t : A path consistency approximation to the minimal labels for Cij i, j = 1,...,n.

2.3.

E a s y ( P o l y n o m i a l T i m e ) S p e c i a l Cases

In this section we explore how far we must restrict the expressive power of the representation language to guarantee t h a t we can compute exact solutions in polynomial time. ValdeVPerez (1987) shows t h a t graphs t h a t are not labeled w i t h disjunctions can be solved exactly in 0 ( n 3 ) time using Allen's algorithm. V i l a i n and K a u t z (1986) claim something stronger. They define a time point algebra for representing and reasoning about the possible relations between points, as opposed to intervals. L e t denote V i l a i n and # Kautz's point algebra. P A is the algebraic structure w i t h underlying set and binary operators intersection and composition. Note t h a t for example, is an abbreviation of and ? means there is no constraint between t w o points, Intersection is then set intersection. Composition is defined as in the interval algebra (equation 2.1) except t h a t multiplication is now given by,

F i g u r e 1 Three and Four Consistency Algorithm A n o t h e r way to view the action of the path consistency a l g o r i t h m together w i t h the definition of composition of labels (equation 2.1) is t h a t entry C i j , the label on the edge ( t , j), gets updated to be the set of elements of the old label t h a t can be p a r t of a consistent singleton labeling of the triangle (i, k, j). T h a t is, we ensure t h a t the labels are minimal w i t h respect to all triangles (or 3-cliques) and composition is defined over labels on edges t h a t share a vertex. The simple idea for i m p r o v i n g the approximation is then to ensure t h a t the labels are minimal w i t h respect to all subgraphs of four vertices (or 4-cliques) and define composition over labels of triangles t h a t share an edge.

The modified path consistency algorithm is given in Figure 1. The algorithm w i t h the new definition of composition (equation 2.2) ensures t h a t C ij becomes the set of elements of the old label t h a t can be part of a consistent singleton labeling of the subgraph of four vertices. If a label on an edge changes it may in t u r n constrain other labels so procedure R E L A T E D P A T H S returns all structures of four vertices in which the edge participates, t a k i n g i n t o account symmetries to prevent redundant c o m p u t a t i o n . Theorem three and

Vilain and K a u t z show t h a t a subset of the interval algebra can be translated into this time point algebra. Let SP # be the set of labels in the interval algebra that can be translated into relations between the endpoints of the intervals using the underlying set of PA#. Vilain and K a u t z assert (Theorem 4, p. 380) that the path consistency algorithm (Allen's algorithm) is exact for computing the minimal labels between points. The consequences for the interval algebra are the following. If their claim is true we can solve the subset SP# of the interval algebra exactly by first translating into the point algebra. However, their claim is false. Here we present a counter-example demonstrating t h a t the path consistency algorithm is not exact for Vilain and Kautz's point algebra. The counter-example also shows t h a t path consistency is not exact for SP* if, instead of first translating into the point algebra, we use the interval algebra representation directly. Below is the interval algebra representation of the example w i t h labels chosen f r o m SP*.

1. The algorithm of Figure 1, achieves A four consistency and requires 0(n ) time.

The idea for developing the i n i t i a l better approximation algorithm can be generalized to develop successively more expensive algorithms t h a t compute progressively better approximations. B u t higher orders of consistency quickly become impractical for all but the smallest problems.

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where

T h e t r a n s l a t i o n i n t o the p o i n t representation i s the f o l l o w i n g (where A - and A + represent the s t a r t a n d end p o i n t s o f i n t e r v a l A , respectively)

T h e o r e m 2 is p r o v e d by s h o w i n g by i n d u c t i o n t h a t if all labels are f r o m SP or PA a n d there is p a t h consistency t h e n the g r a p h is s t r o n g l y Ar-consistent for all k < n. Hence t h e labels are exact. The inductive step relies on t h e f a c t t h a t t h e relations between vertices (the labels) f o r m convex sets a l l o w i n g the applicat i o n of L e m m a 1. T h e p r o o f is c o n s t r u c t i v e and gives an a l g o r i t h m for f i n d i n g consistent singleton labelings. In t h e r e m a i n d e r of this section we show t h a t the f o u r consistency a l g o r i t h m developed in section 2.2 is exact for SP* and PA* (recall PA* includes #, PA doesn't). If 5* is p e r m i t t e d in t h e language of the p o i n t algebra, the relations no longer f o r m convex sets a n d p a t h consistency is no longer sufficient. Here is the smallest counter-example to the exactness of p a t h consistency for PA* a n d , up to i s o m o r p h i s m , is the only counter-example of f o u r vertices.

A p p l y i n g a p a t h consistency a l g o r i t h m results in no changes; the relations between p o i n t s are all considered to be m i n i m a l . H o w e v e r , the r e l a t i o n A- < B i s n o t m i n i m a l , t h u s d e m o n s t r a t i n g t h a t the a l g o r i t h m is n o t exact f o r t h e p o i n t algebra. T h e m i n i m a l relat i o n is A- < B + . T h i s change is also reflected in the o r i g i n a l i n t e r v a l algebra r e p r e s e n t a t i o n : the m i n i m a l label between v e r t e x A a n d v e r t e x B is { d , d i , o, o i , f, f i } , w i t h the meets r e l a t i o n h a v i n g been d r o p p e d because it could n o t p a r t i c i p a t e in any consistent singleton labeling o f the g r a p h . Interestingly, Allen's a l g o r i t h m is also n o t exact w h e n applied to the i n t e r v a l algebra representation of this example, whereas the a l g o r i t h m we proposed in the previous section c o m putes the m i n i m a l labeling. T o r e i t e r a t e , the counterexample shows t h a t the p a t h consistency a l g o r i t h m is n o t exact for SP* a n d PA*. We have been i n f o r m e d t h a t L a d k i n (1989) shows t h a t for PA* p a t h consistency does guarantee t h a t , if the m i n i m a l labels on edges is the e m p t y set, this w i l l be detected. Define a new p o i n t algebra, PA, w i t h the same b i n a r y operators and u n d e r l y i n g set as PA* w i t h the exception t h a t ^ is excluded f r o m the u n d e r l y i n g set. L e t SP be the set of labels in the i n t e r v a l algebra t h a t can be t r a n s l a t e d i n t o relations between the endpoints of the intervals using the u n d e r l y i n g set of PA. P a t h consistency is exact for SP and for PA. T h e f o l l o w i n g l e m m a on the intersection of convex sets is useful in the p r o o f of exactness. L e m m a 1 (Kelly's t h e o r e m ) . Let F be a finite family of at least n-hl convex sets in Rn such that every n+1 sets in F have a point in common. Then all the sets in F have a point in common. T h e o r e m 2. The path consistency algorithm is exact if all labels are chosen from SP. It is also exact for PA, the point algebra that excludes the ^ relation.

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T h e g r a p h is p a t h consistent. B u t this is n o t the m i n i m a l labeling since n o t every element is capable of being p a r t of a consistent singleton labeling. To see t h i s , choose t h e singleton labeling f o r the t r i a n g l e ( A , B, C) such t h a t A = B = C. We now have D < A, B < D, a n d D ^ C. U s i n g s t a n d a r d i n t e r v a l n o t a t i o n and s u b s t i t u t i o n of equals, D m u s t be in the three i n t e r v a l s (—00, a ] , [ a , +00), a n d (—00, a) U ( a , +00). It is easily seen t h a t these i n t e r v a l s are pair-wise consistent b u t together are inconsistent. If the g r a p h was also made f o u r consistent, say by a p p l y i n g a l g o r i t h m A114 (Figure 1), the label between A and B w o u l d be > and this counter-example c o u l d n o t occur. T h e counter-example t h e n is unique for n = 4 and c a n n o t occur if the g r a p h is three a n d f o u r consistent. B u t can we f i n d a c o u n t e r - e x a m p l e for n > 4? T h e answer is, no. It can be s h o w n (see the longer version of the paper) t h a t any larger counter-example must have a s u b g r a p h of 4 vertices isomorphic to the example above. B u t this is r u l e d o u t by f o u r consistency. T h e o r e m 3. Hie three and four consistency algorithm of Figure 1 is exact if all labels are chosen from SP*. It is also exact for PA*, the time point algebra that includes the 5* relation. We characterized the subsets of the i n t e r v a l algebra for w h i c h the p a t h consistency a l g o r i t h m and the four-consistency a l g o r i t h m are exact. Unfortunately these subsets are s m a l l . We m u s t quite severely rest r i c t o u r representation language t o guarantee efficient and exact solutions.

3.

The One-to-All Problem

The algorithms given in the previous section compute approximations to the m i n i m a l labels between every interval and every other interval (the all-to-all version of the problem). If we are only interested in the relationships between one i n t e r v a l and every other interval or between t w o particular intervals then, in computing the relationships between all intervals, we may be doing too much w o r k . In this section we present an efficient a l g o r i t h m for the one-to-all version of the problem and show t h a t the algorithm is exact for a useful subset of the i n t e r v a l and point algebras.

I n p u t : A source vertex s and a matrix C where element C ij , is the label on edge ( i , j) O u t p u t : An approximation to the minimal labels for Csj, j = 1, . ,n

3.1. A One-to-All Approximation Algorithm O u r solution to the one-to-all version of the problem is an adaptation of D i j k s t r a ' s (1959) algorithm for comp u t i n g the shortest p a t h f r o m a single source vertex to every other vertex. His a l g o r i t h m , which can be categorized as a label-setting a l g o r i t h m , produces poor quality solutions when applied to the interval algebra. In the a l g o r i t h m of Figure 2, we allow a label to change after it has been tentatively fixed and perhaps f u r t h e r constrain other labels. This change turns the algorithm i n t o a label-correcting algorithm where no labels are considered final u n t i l the procedure halts. This change to D i j k s t r a ' s algorithm also appears in Edmonds and K a r p (1972) in the context of finding shortest paths where negative arc lengths are allowed. Johnson (1973) showed t h a t , if the labels are integers, this change makes the algorithm exponential in the worst case. In this context, t h o u g h , the algorithm is 2

0(n ). T h e o r e m 4. requires 0(n2)

The time.

one-to-all

algorithm

of Figure

2

3 . 2 . E a s y ( P o l y n o m i a l T i m e ) S p e c i a l Cases Here we explore how far we must restrict the expressive power of the representation language to guarantee t h a t our one-to-all approximation algorithm of Figure 2 (One3) is exact. N o t e t h a t the all-to-all algorithms compute approximations to all the minimal labels, but even the labels we are not interested in help us by f u r t h e r constraining the labels we are interested i n . One3 does not do this; it uses less information to compute its approximations. Hence, in general its approximations are poorer t h a n those of the all-to-all algor i t h m s . Surprisingly t h o u g h , One3 is exact for the same subset of the interval algebra for which the path consistency (Allen's algorithm) is exact. T h e o r e m 5. The label-correcting algorithm of Figure 2 is exact for SP and PA, provided the minimal label on an edge is not the empty set or null relation. The proof of the theorem uses the property t h a t composition distributes over intersection. This property is true for SP and PA only if we can guarantee t h a t the intersection of t w o labels w i l l never result in the empty set. It is easy to show t h a t the theorem is false if the empty set is the m i n i m a l label on an edge.

4. E x p e r i m e n t a l Results In this section we present the results of some computational experiments comparing the quality of the solutions produced by Allen's and our approximation algorithms. The experiments give a partial answer to the question: w i t h what degree of confidence can we rely on the less expensive approximate solutions? We also present a simple test for predicting when the approximation algorithms w i l l and w i l l not produce good quality approximations. For each problem of size n we randomly generated a consistent singleton labeling and then added uncertainty in the form of additional disjuncts on the possible relationships between t w o intervals. We then applied the three approximation algorithms, chose a particular edge, determined the minimal or exact label on that edge using an exact backtracking algorithm, and recorded whether the less expensive approximate solutions differed from the exact solution. Table 1 summarizes the results for t w o distributions and three algorithms: One3 (Figure 2), A113 (the path consistency algorithm), and A114 (Figure 1). Distribution one was chosen to roughly approximate instances that may arise in a planning application (as estimated f r o m a block-stacking example in Allen and Koomen, 1983). Fortunately, for this class of problems the results suggest t h a t for a reassuringly large percentage of the time we can use the path consistency algorithm w i t h near impunity: the outcome is the same as t h a t of using an exact algorithm. W i t h a different distribution, however, up to two-thirds of the labels on average were not minimal. Let SP be the subset of the interval algebra discussed earlier t h a t can be solved exactly using the path consistency algorithm. Computational evidence shows a strong correlation between the percentage of the t o t a l labels t h a t are f r o m SP and how well the One3, A113, and A114 algorithms approximate the exact solution. Recall t h a t Theorem 2 (Theorem 5) states

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Table 1 A v e r a g e p e r c e n t a g e differences b e t w e e n t h e a p p r o x i m a t i o n a l g o r i t h m s a n d a n exact a l g o r i t h m f o r v a r i o u s p r o b l e m sizes. Distribution 1: About 75% of the time the uncertainty added is {1} and the remaining time consists of from 0 to 3 of the basic relations. Distribution 2: A l l labels are equally likely to be added as uncertainty. 150 tests were performed for each problem size, n

A l l e n , J . F., and J . A . K o o m e n . 1983. P l a n n i n g Using a Temporal World Model. Proceedings of the Eighth International Joint Conference on Artificial Intelligence (IJCAI), K a r l s r u h e , W. G e r m a n y , 741-747. D i j k s t r a , E . W . 1959. A N o t e o n T w o Problems i n Connexion w i t h Graphs. Numerische Mathematik 1 , 269-271. E d m o n d s , J . , and R . M . K a r p . 1972. T h e o r e t i c a l I m p r o v e m e n t s i n A l g o r i t h m i c E f f i c i e n c y for N e t w o r k F l o w P r o b l e m s . J. ACM 1 9 , 248-264. F r e u d e r , E. C. 1978. Synthesizing C o n s t r a i n t Expressions. Comm. ACM 2 1 , 958-966. F r e u d e r , E. C. 1982. A Sufficient C o n d i t i o n B a c k t r a c k - F r e e Search. J. ACM 2 9 , 24-32.

t h a t A113 (One3) is exact w h e n all the labels are f r o m SP so we c a n n o t i m p r o v e on t h a t . B u t , as the percentage of t h e t o t a l labels t h a t are f r o m SP nears zero, an increasing n u m b e r of the labels (on average) assigned b y the a p p r o x i m a t i o n a l g o r i t h m s are n o t m i n i m a l . T h u s we have an effective test for p r e d i c t i n g w h e t h e r it w o u l d be useful to apply a more expensive algorithm.

5.

Conclusions

We considered a p o p u l a r r e p r e s e n t a t i o n for t e m p o r a l relationships between i n t e r v a l s i n t r o d u c e d by James A l l e n a n d its associated c o m p u t a t i o n a l or reasoning p r o b l e m of, given possibly i n d e f i n i t e knowledge of the relations between some i n t e r v a l s , c o m p u t i n g the strongest possible assertion about the relations between some or all i n t e r v a l s . A l l e n gives an a p p r o x i m a t i o n a l g o r i t h m based o n c o n s t r a i n t p r o p a g a t i o n . W e presented a n a l g o r i t h m for c o m p u t i n g b e t t e r a p p r o x i m a t i o n s for the a l l - t o - a l l version of the p r o b l e m and a test for p r e d i c t i n g w h e n this more expensive a l g o r i t h m is useful. We presented an a l g o r i t h m for the one-to-all version of the p r o b l e m and a test for p r e d i c t i n g w h e n this less expensive a l g o r i t h m is useful. We gave a c o u n t e r example to a result in the l i t e r a t u r e and i d e n t i f i e d easy ( p o l y n o m i a l t i m e ) special cases of b o t h versions o f t h e p r o b l e m .

Acknowledgements I t h a n k R o b i n C o h e n , B r u c e Spencer, Fei Song, F a h i e m Bacchus, A n d r e T r u d e l , and P a u l v a n A r r a g o n for help, advice, and encouragement. F i n a n c i a l supp o r t was g r a t e f u l l y received f r o m the N a t u r a l Sciences and E n g i n e e r i n g Research C o u n c i l o f C a n a d a , I T R C , and the U n i v e r s i t y o f W a t e r l o o .

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