Bayardo, R. J., and Miranker, D. P. 1996. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In AAAI-96, 298--304.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Theorem 4 Let m d be the depth of a DFS tree traversal of the induced graph 25 G d of G. If d is a DFS ordering of G d , then the complexity of graph based backjumping using ordering d is O(exp(m d ) A proof, that uses somewhat di erent terminology and derivation, is given in [5]. The virtue of Theorem 4 is in allowing a larger set of orderings, each yielding a bound on backjumping s performance as a function of its DFS tree depth, to be considered. Since every DFS ordering of G is also a DFS ordering of its induced graph along d, G d (the added induced arcs are back ....

....bounded learning algorithm has an overhead complexity that is time and space exponentially bounded by i. An alternative to bounding the size of learned nogoods is to bound the learning process by discarding nogoods that appear to be no longer relevant, by some measure. De nition 13 (i relevant) [5] A nogood is i relevant if it di ers from the current partial assignment by at most i variable value pairs. De nition 14 (i th order relevance bounded learning) 5] An i th order relevance bounded learning scheme maintains only those learned nogoods that are i relevant. The dynamic backtracking ....

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R. Bayardo and D. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In AAAI-96: Proceedings of the Thirteenth National Conference on Arti cial Intelligence, pages 298-304, Portland, OR, 1996.

....two variables x i (i # j 1 . j k such that the current value of x i is di#erent from v i . Notice that if we generalizes to not only constraint having at least two such variables but at least k 1 such variables, one obtain a well known generalization of DBT: k relevance bounded learning [1]. The second di#erence with CBJ is the repair operator of the M component, which consists here in only erasing the most recent culprit decision constraint. where C # D is CD where c is the latest decision constraint such that C # D is compatible with CL (by construction, we know ....

Roberto J. Bayardo Jr. and Daniel P. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In AAAI'96, 1996.

....user friendly solutions to constraint problems, 10] generates tree like explanations and combines them with ordering heuristics and selection strategies to obtain better explanations according to a well defined criterion, etc. In this paper, we advocate for the use of k relevant explanations [11]. The idea is to record bounded sets of explanations [13] rather than a single one) based on relevance (how far is the validity of the explanation considering the current situation this is evaluated with k) k relevant explanations, by providing a relevance based long term memory for ....

....to a given value n. This criterion limits the spatial complexity, but may forget really interesting conflict sets. Relevance bounded criterion: explanations are kept if they are not too far from the current set of decision constraints. This concept (called k relevance) has been introduced in [11] and focus explanations conflict set management to what is important: relevance w.r.t. the current situation. Time and size bounded recording do have a controllable space complexity. This is also the case for k relevance learning (cf. Section 3) As we shall see, our tools are meant for the ....

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Bayardo Jr., R.J., Miranker, D.P.: A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In: AAAI'96. (1996)

....variable assignment. By doing so, the space complexity remains polynomial. A contradiction explanation that does not contain such a constraint denotes an overconstrained problem. A nogood is said to be relevant if all the decision constraints in it are still valid in the current search state [2]. Explanations can be used in several ways [13, 11, 14] Debugging purposes pop to the mind: to explain clearly failures, to explain di#erences between intended and observed behavior for a given problem (e.g. why is value 4 not assigned to variable x ) Explanations can also be used to ....

Roberto J. Bayardo Jr. and Daniel P. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In AAAI'96, 1996.

....problems, 6] and [7] introduce tools to dynamically remove constraints, PALM [8] uses conflicts to address those issues and define new search algorithms, etc. User interaction requires user friendly and interactive tools. In this paper, we advocate for the use of k relevant explanations [9] to provide the COINS (COnstraint based INteractive Solving) system. COINS helps the user understand inconsistency, simulate constraint additions and or retractions (without any propagation) determine if a given constraint belongs to a conflict and provide diagnosis tools (e.g. why variable v ....

....or equal to a given value n. This criteria limits the spatial complexity, but may forget really interesting nogoods. Relevance bounded criterion: explanations are kept if they are not too far from the current set of decision constraints. This concept (called k relevance) has been introduced in [9] and focus explanations conflict management to what is important: relevance wrt the current situation. Time and size bounded recording do have a controllable space complexity. This is also the case for k relevance learning (cf. section 3) As we shall see, our tools are meant for the debugging and ....

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Bayardo Jr., R.J., Miranker, D.P.: A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In: AAAI'96. (1996)

....elimination tolerance and preserving at least one solution, are the same basic requirements as for using correct procedures. The third condition turns out to be satisfied by almost any systematic engine, in particular depth first search, limited discrepancy search, relevance bounded backtracking [20], size bounded backtracking and dynamic backtracking all satisfy this condition. 3.4 Overview of Correctness Results In this chapter we have provided a set of conditions that have been proven to guarantee systematicity and completeness for search engines using procedural reasoning under a ....

Roberto J. Bayardo Jr. and Daniel P. Miranker. A complexity analysis of spacebounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, pages 298--304, 1996.

....relevance to the current position in the search space. The idea originated in dynamic backtracking [17] That algorithm deletes a nogood when one or more variable assignment pairs are no longer a member of the current partial assignment. Bayardo and Miranker defined a generalized form of relevance [2] in which a nogood is i relevant if it di#ers from the partial assignment in at most i variables. Keeping all nogoods that are i relevant is called i relevance bounded learning. We can view the relevance measure of a clause as the number of variables we must unbind before the clause can be ....

....true, b = true, c = true, d = true , the nogood (7) will be 0 relevant and the nogood (6) will be 4relevant. This latter nogood will therefore be dropped because it has exceeded the relevance bound of 3. Dropping clauses that exceed the relevance bound enables us to maintain polynomial space usage [2]. Experimentally, relevancebounded learning makes better use of space resources than does k order learning, and even when relevance bounded learning is restricted to linear space, the performance is comparable to that of unrestricted learning [2] Both relevancebounded and k order learning are far ....

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R. J. Bayardo and D. P. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, pages 298-- 304, 1996.

....is only recorded if it contains at most i variables. The technique is extended to also record nogoods at internal nodes in [1] Ginsberg [3] proposes instead to record all nogoods, but discard them once they are no longer relevant to the current branch of the search tree. Bayardo and Miranker [4] generalize that proposal to i order relevance bounded nogood recording, in which a nogood is discarded as soon as it differs from the current branch in i variables. Moskewicz et al. [5] however, have shown that the cpu cost of nogood recording can be alleviated. They introduce the notion of ....

R. Bayardo Jr. and D. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, pages 298--304, Portland, Oregon, 1996.

....relevance to the current position in the search space. The idea originated in dynamic backtracking [17] That algorithm deletes a nogood when one or more variable assignment pairs are no longer a member of the current partial assignment. Bayardo and Miranker de ned a generalized form of relevance [2] in which a nogood is i relevant if it di ers from the partial assignment in at most i variables. Keeping all nogoods that are i relevant is called i relevance bounded learning. We can view the relevance measure of a clause as the number of variables we must unbind before the clause can be ....

....true; b = true; c = true; d = trueg, the nogood (7) will be 0 relevant and the nogood (6) will be 4relevant. This latter nogood will therefore be dropped because it has exceeded the relevance bound of 3. Dropping clauses that exceed the relevance bound enables us to maintain polynomial space usage [2]. Experimentally, relevancebounded learning makes better use of space resources than does k order learning, and even when relevance bounded learning is restricted to linear space, the performance is comparable to that of unrestricted learning [2] Both relevancebounded and k order learning are far ....

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R. J. Bayardo and D. P. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Arti cial Intelligence, pages 298{ 304, 1996.

..... When identifying a dead end while assigning v i , cbj considers the most recent variable in CS v i to be a culprit. But 2 The nogood is a logical consequence of the set of constraints C. 3 A nogood is said to be relevant if all the assignments in it are still valid in the current search state [3]. 4 as opposed to dbt, with cbj a backtrack occurs: the conflict sets and domains of the future variables are reset to their original value. By doing so, cbj forgets a lot of information that could have been useful. This leads to thrashing. dbt, similarly to cbj, selects the most recent ....

Roberto J. Bayardo Jr. and Daniel P. Miranker. A complexity analysis of spacebounded learning algorithms for the constraint satisfaction problem. In AAAI'96, 1996.

....systems: To explain why no solution is found to a given problem. As stated before, a contradiction explanation that does not contain any decision constraints denotes an 3 A nogood is said to be relevant if all the decision constraints in it are still valid in the current search state [Bayardo Jr. and Miranker, 1996] . over constrained system (i.e. a system with no possible solutions) Such explanations are recursively obtained after having tested all possible values for a given variable. The interested reader should refer to [Jussien and Barichard, 2000] for more information. To provide a data driven ....

Roberto J. Bayardo Jr. and Daniel P. Miranker. A complexity analysis of spacebounded learning algorithms for the constraint satisfaction problem. In AAAI'96, 1996.

....Dependency based methods have been of limited use in practice because of the overhead involved in constructing and using the collection of accumulated reasons. This problem has been substantially addressed in the work on dynamic backtracking (Ginsberg, 1993) and its successors such as relsat (Bayardo Miranker, 1996), where polynomial limits are placed on the number of nogoods being maintained. In game search, however, most algorithms already include signi cant cached information in the form of a transposition table (Greenblatt, Eastlake, Crocker, 1967; Marsland, 1986) A transposition table stores a ....

Bayardo, R. J., & Miranker, D. P. (1996). A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Articial Intelligence, pp. 298-304.

....jump back learning, it uses a different method to limit the size of the nogood set. In Tableau CBJ 3 rd Lrn only those no goods involving at most 3 variables are stored. In relsat all no goods are stored but the set is reduced during search by relevance bounding [Ginsberg McAllester, 1994; Bayardo Miranker, 1996]. When a no good differs in more than a fixed number of assignments from the current partial label, it is removed from the set. The idea here is that as the assignments in the current label diverge more and more from the no good, the no good becomes less and less relevant in pruning search. ### ....

R. J. Bayardo and D. Miranker, A complexity analysis of space bounded learning algorithms for the constraint satisfaction problem, Proceedings AAAI-96, 1996

....of known hard problem instances from timetabling, planning, code optimization, and circuit synthesis. We also considered CSP encodings of scheduling problems. For our solvers, we used two state of the art satisfiability engines: Satz by Li and Anbulagan [39] and Relsat by Bayardo and Schrag [5], and an e#cient CSP solver built using the Ilog C constraint programming library [53] It is important to note that the underlying deterministic complete search engines are among the fastest (and on many problems, the fastest) in their class. Thus, the techniques discussed in this paper extend ....

....strategies, such as conflict 2 We thank Chu Min Li and Roberto Bayardo for making their source code available to us. See also SATLIB at www.informatik.tu darmstadt.de AI SATLIB . heavytails.tex; 4 10 1999; 23:15; p. 5 6 Gomes et al. directed backjumping (CBJ, 52] and relevance bounded learning [5]. These procedures are the fastest SAT methods we have found for the instances discussed in this paper. In both procedures the powerful heuristics often only yield a relatively small set of variables to branch on. We therefore added the heuristic equivalence parameter H to enlarge the choice set. ....

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Bayardo, R. and D. Miranker: 1996, `A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem'. In: Proceedings of the Thirteenth National Conference on Artificial Intelligence (AAAI-96). Portland, OR, pp. 558--562.

....violation test. If the new nogood includes an unknown variable, the agent has to request the corresponding agent to send its value. 3 Nogood Learning Constraint satisfaction algorithms can be enhanced by look back techniques, which exploit information about search that has already been done [2, 3, 5, 8, 9, 10, 11, 19] This paper provides a new learning method for a distributed constraint satisfaction algorithm that is based on a lookback technique in the CSP literature. 3.1 Resolvent based Learning We use a similar method to that presented in [5, 10, 11, 19] This method can be summarized as follows: for ....

....that are relevant to its variable, so the nogoodexplosion for each agent is not so serious. 5 However, we cannot say that the AWC with nogood learning is completely free of the nogood explosion. In the CSP literature, two approaches, size bounded learning [9, 10] and relevance bounded learning [2] have been proposed to handle the nogood explosion. The sizebounded learning is a simple strategy that bounds the size of the recorded nogood. The relevance bounded learning, on the other hand, records nogoods of arbitrary size, but only maintains i relevant nogoods, i.e. nogoods that differ from ....

Bayardo, R. J. and Miranker, D. P.: A Complexity Analysis of Space-Bounded Learning Algorithms for the Constraint Satisfaction Problem. Proceedings of the Thirteenth National Conference on Artificial Intelligence (1996) 298-- 304

....on the other. Storing the failure explanations and search control rules which are learned from all the interior nodes could be very expensive. The CSP and machine learning literatures took different approaches to this problem. Researchers in CSPs (e.g. Detcher 1990; Frost Dechter 1994a] Bayardo Miranker 1996]) concentrated on the syntactic characteristics of the failure explanations, such as their size and minimality, to decide whether or not they should be stored. Researcher in machine learning concentrated instead on utility analysis approaches that use the distribution of the encountered problems ....

Bayardo, R. J. and Miranker, D. P., A Complexity Analysis of Space-Bounded Learning Algorithms for the Constraint Satisfaction Problem, AAAI96, pp. 298-304, 1996

....end before the date 7. CLAIRE run time library v 2.5.1 [os: ntw, C :MSVC ] CLAIRE interpreter Copyright (C) 1994 97 Y. Caseau (see about( Choco version 0.23, Copyright (C) 1999 2000 F. Laburthe Palm version 0.2.10, Copyright (C) 2000 N. Jussien palm shedulingProblem( eval[1] f: 7. 15] our project cannot end before date 7 palm Fig. 2. Answer to a simple scheduling problem The PaLM system provides much more than a simple propagation mechanism: it is able to provide explanations for basic events. Suppose that we want to end the project at date 6. In a classical ....

....= 1) then the upper bound of the second variable (c.v2) should be updated too (c.idx2 gives the index of constraint c in variable c.v2) The corresponding PaLM code is given on figure 7. 4 A nogood is said to be relevant if all the assignments in it are still valid in the current search state [1]. The original choco variable IntVar : AbstractVar( inf:integer, sup:integer, value:integer = unknown ) The inherited PaLM variable PalmIntVar : IntVar( explanationOnInf:list[PalmExplanation] nil, storing structure explanationOnSup:list[PalmExplanation] nil ) Fig. 5. A ....

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Roberto J. Bayardo Jr. and Daniel P. Miranker. A complexity analysis of spacebounded learning algorithms for the constraint satisfaction problem. In AAAI'96, 1996.

....end before the date 7. CLAIRE run time library v 2.5.1 [os: ntw, C :MSVC ] CLAIRE interpreter Copyright (C) 1994 97 Y. Caseau (see about( Choco version 0.23, Copyright (C) 1999 2000 F. Laburthe Palm version 0.2.10, Copyright (C) 2000 N. Jussien palm shedulingProblem( eval[1] f: 7. 15] our project cannot end before date 7 palm Fig. 2. Answer to a simple scheduling problem The PaLM system provides much more than a simple propagation mechanism: it is able to provide explanations for basic events. Suppose that wewantto end the project at date 6. In a classical ....

....idx = 1)thenthe upper bound of the second variable (c.v2) should be updated too (c.idx2 gives the index of constraint c in variable c.v2) The corresponding PaLM code is given on figure 7. 4 A nogood is said to be relevant if all the assignments in it are still valid in the current search state [1]. 6 The original choco variable IntVar : AbstractVar( inf:integer, sup:integer, value:integer = unknown ) The inherited PaLM variable PalmIntVar : IntVar( explanationOnInf:list[PalmExplanation] nil, storing structure explanationOnSup:list[PalmExplanation] nil ) Fig. 5. ....

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Roberto J. Bayardo Jr. and Daniel P. Miranker. A complexity analysis of spacebounded learning algorithms for the constraint satisfaction problem. In AAAI'96, 1996. 13

....trees have the property that adjacent vertices from the original graph reside in the same branch of the tree. So DFS trees are pseudo trees but a pseudo tree is not necessarily a DFS tree of G (it may include edges with are not in G) This pseudo tree notion has been further studied in [BM95, BM96] where it is mentioned that pseudo tree are also known as rootedtree arrangement [Gav77] Note however that optimal rooted tree arrangements and minimum height pseudo tree are not a priori equivalent since the criteria applied to rooted tree arrangement (see [GJ79] page 201) is the sum of the ....

Roberto Bayardo and Daniel Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proc. of AAAI'96, pages 298--304, Portland, OR, 1996.

....the schedule in a single pass (in this case the distributed mistake is the failure to put off tasks that can be done in the later night shifts) This complements LDS ability to fix more subtle spot mistakes in the heuristic. The third mature technique is now called relevance bounded learning [2] in the literature, but we often refer to it as dynamic backtracking light. One lesson we 118 have learned over the course of this project is that dynamic backtracking differs from previous algorithms in two orthogonal directions. First, like k bounded learning or dependency directed ....

R. Bayardo and D. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, 1996.

....the cost of an assignment grows according to nogood recording. Finally, even if the original problem is a binary one (i.e. involving only binary constraints) nogood storage can add higher arity constraints, which are source of expensive communication, expensive checking, and memory explosion [3, 1]. Our goal in this section is the evaluation of our method in a really distributed environment, when it is implemented with and without the distributed variable ordering algorithm of Section 3.1. We gave a physical processor to each agent in the multi agents system and a variable to each agent. ....

R.J. Bayardo Jr. and D.P. Miranker, `A complexity analysis of spacebounded learning algorithms for the constraint satisfaction problem', in Proceedings AAAI, pp. 298--304, (1996).

....work on finding practical algorithms for solving SAT. There are powerful systematic solvers based on the classical DavisPutnam (DP) procedure [14, 15] but incorporating new heuristics, intelligent backtracking, and learning mechanisms. Recent examples are posit[21] ntab [13] satz [42] and relsat [2]. There are also non systematic algorithms based on local search: Some examples are the Breakout method of Morris [49] gsat [66, 68] and wsat [67] A major problem with using such solvers is that encoding natural problems into SAT results in very large formulas. We will see examples in which ....

....store entailed clauses as they proceed. The main problem is that if we store all the clauses memory usage grows with linearly with runtime, and this gives us too many clauses. There have been two main proposals to limit the memory usage: k order learning [16] and Relevance Bounded Learning (RBL) [2, 3]. In k order learning we only store clauses of length k or less. RBL uses a much more selective criterion that adapts to the portion of the search space currently under exploration. The key idea comes from dynamic backtracking [24, 25] and is to limit the number of literals in a clause that do ....

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Roberto J. Bayardo and Daniel P. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence (AAAI--96), pages 298--304, Portland, OR, 1996.

....duration between adjacent tokens are not represented in this figure. This representation has some subtle but important ramifications for sophisticated CSP algorithms. Consider, for example, powerful no good learning techniques employed by algorithms such as Dynamic Backtracking [Gin93] RelSat [BM96] and Tabu search [Glo89] A no good is simply a combination of variable assignments which cannot be part of a solution. No goods containing values from dynamic domains are, unfortunately, no good when the value changes during search. To see why, consider a no good containing a token insertion ....

R. Bayardo and D. Miranker. A complexity analysis of space bounded learning algorithms for the constraint satisfaction problem. Proceedings of the 13th National Conference on Artificial Intelligence, pages 298--304, 1996.

....Theorem 5 Let m d be the depth of a DFS tree traversal of some induced graph G . The complexity of graph based backjumping using ordering d of a constraint problem having a constraint graph G, is O(exp(m d ) A proof, that uses somewhat different terminology and derivation, is given in [BM96]. The virtue of the above Theorem is in allowing a larger set of orderings, each yielding a bound on backjumping s performance as a function of its DFS tree depth, to be considered. Since it can be shown that every DFS traversal of G is also a DFS traveral of its induced graph along d, G d , ....

....lines in Figure 11 will be generated by the former but not by the latter. 5.5 Relevance bounded learning An alternative to bounding the size of learned nogoods is to bound the learning process by discarding nogoods that appear to be no longer relevant, by some measure. Definition 12 (i relevant) [BM96] A nogood is i relevant if it differs from the current partial assignment by at most i variable value pairs. Definition 13 (i th order relevance bounded learning) BM96] An i th order relevance bounded learning scheme maintains only those learned nogoods that are i relevant. Ginsberg s dynamic ....

[Article contains additional citation context not shown here]

R. Bayardo and D. Miranker. A complexity analysis of spacebounded learning algorithms for the constraint satisfaction problem. In AAAI-96: Proceedings of the Thirteenth National Conference on Artificial Intelligence, pages 298--304, Portland, OR, 1996.

....a reduced cost bound in the function being optimized. An optimal solution is found by determining the lowest cost bound for which the corresponding constraint satisfaction problem has a solution. A similar approach was used recently to find a shortest plan using satisfiability and CSP techniques [15, 2]. We present experiments with five algorithms that have proven most useful when tested on random problems. In general, when an algorithm is applied to a maintenance problem instance, it solves each of the corresponding CSPs independently. For the new iterative learning procedure, an algorithm ....

Roberto Bayardo and Daniel Mirankar. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, pages 298--304, 1996.

....at solving decision problems such as planning, even in realworld domains. Among the many candidate search techniques that may prove applicable to this planning framework are dependency directed search (Stallman Sussman 1977) limited discrepancy search (Harvey 1995) relevance bounded search (Bayardo Jr. Miranker 1996), iterative sampling (Langley 1992) heuristic biased sampling (Bresina 1996) and repairbased search (Minton et al. 1990) 5. THE CONSTRAINT REASONING SYSTEM The Next Generation Remote Agent Planner is based on a redesign of the existing RA planner and thus inherits a number of existing solutions ....

Bayardo Jr., R. J. & D. P. Miranker (1996). A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, pages 298--304.

....can be of substantial benefit, especially on hard constraint satisfaction problems. Of course, the LVO heuristic does not always correctly predict which values will lead to solutions, but it is frequently more accurate than an uninformed ordering of values. We refer to BJ DVO with learning [4, 9, 2] as BJ DVO LRN. Learning in CSPs, also known as constraint recording, involves a during search transformation of the problem representation into one that may be searched more effectively. This is done by enriching the problem description by new constraints, also called no goods, which do not ....

Roberto Bayardo and Daniel Mirankar. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, pages 298--304, 1996.

....known on the larger problems have been found by using swo to generate candidate solutions and then refining these solutions using OR techniques. 5 Relevance bounded learning 5. 1 Method The final approach we will discuss to the early mistake problem is known as relevance bounded learning (rbl) [1, 2]. The basic idea is an extension of truth maintenance [7, 9, 10] In a truth maintenance system, a reason (or nogood ) is recorded when one is forced to backtrack during the search process. The basic difficulty with this is that search techniques spend the bulk of their time backtracking and it ....

R. J. Bayardo and D. P. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, pages 298--304, 1996.

....a reduced cost bound in the function being optimized. An optimal solution is found by determining the lowest cost bound for which the corresponding constraint satisfaction problem has a solution. A similar approach was used recently to find a shortest plan using satisfiability and CSP techniques [12, 2]. We present experiments with five algorithms that have proven most useful when tested on random problems. In general, when an algorithm is applied to a maintenance problem instance, it solves each of the corresponding CSPs independently. For the new iterative learning procedure, an algorithm ....

Roberto Bayardo and Daniel Mirankar. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, pages 298--304, 1996.

....subproblems. Instances which have a large number of solutions, like those in Table 1, tend to have only few such subproblems, and these are identified early on by the ordering optimizations that prevent futile counting when unsatisfiable subproblems are present. Complexity Related Issues In [Bayardo Miranker 1996], the constraint graph of a constraint satisfaction problem (CSP of which SAT is a restriction) is recursively decomposed to form a rooted tree arrangement (see figure above) on which a backtracking algorithm similar to DDP is applied. By definition of a rooted tree arrangement, two variables ....

....count of the subproblem is greater than zero) the importance of which is indicated by complexity results. Previous work has shown that recording only nogoods during backtrack search leads to effective structure based bounds on runtime for determining satisfiability [Frost Dechter 1994] Bayardo Miranker [1996] demonstrated that while good recording improves runtime complexity of satisfiability checking, it does so by reducing only the base of the exponent. When counting models, good learning is in fact a necessary enhancement of backtrack search for achieving polynomial time complexity given certain ....

Bayardo, R. J. and Miranker, D. P. 1996. A Complexity Analysis of Space-Bounded Learning Algorithms for the Constraint Satisfaction Problem. In Proc. 13th Nat'l Conf. on Artificial Intelligence, 558-562.

....(branch points) Failure Rate ( 50000] Tab CBJ lrn 78 0 Table 4. Exceptionally hard problem statistics for n=200, m=700, n =50, m =180 Algorithm Mean Difficulty of Solved Instances (branch points) Failure Rate ( 50000] Tab CBJ lrn 3,702 3. 4945 limits on derived clauses [2,16], or some method for efficiently producing smaller cul prit sets than conflict directed backjumping (e.g. possibly along the lines of Dechter s deep learning schemes [9] 6 Related and Future Work Ginsberg McAllester [16] evaluate a CSP algorithm they call partial order dynamic ....

R. J. Bayardo, A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem, In Proceedings of the 13th National Conference on Artificial Intelligence (AAAI-96), 1996.

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Bayardo, R. J. and Miranker, D. P. 1996. A Complexity Analysis of Space-Bounded Learning Algorithms for the Constraint Satisfaction Problem. In Proc. 13th Nat'l Conf. on Artificial Intelligence, 558-562.

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Bayardo, R. J. and Miranker, D. P. 1996. A Complexity Analysis of Space-Bounded Learning Algorithms for the Constraint Satisfaction Problem. In Proc. 13th Nat'l Conf. on Artificial Intelligence, 558-562.

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Bayardo, R. J., and Miranker, D. P. 1996. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In AAAI-96, 298--304.

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Bayardo, R. and D. Miranker: 1996, `A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem'. In: Proceedings 22 of the Thirteenth National Conference on Artificial Intelligence (AAAI-96). Portland, OR, pp. 558--562.

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R.J. Bayardo and D.P. Miranker, `A complexity analysis of spacebounded learning algorithms for the constraint satisfaction problem', in Proc. AAAI 1996.

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R. J. Bayardo, Jr. and D. P. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proc. AAAI, pages 298--304, 1996.

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R. Bayardo and D. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proc. AAAI-96, 1996.

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R. Bayardo and D. Miranker. A Complexity Analysis of Space-Bounded Learning Algorithms for the Constraint Satisfaction Problem. In Proc. of the 34 13th Nat'l Conf. on Arti cial Intelligence, 1996.

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R. J. Bayardo Jr and D. P. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, pages 298--304, Portland, Oregon, 1996.

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R. J. Bayardo Jr and D. P. Miranker. A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, pages 298--304, Portland, Oregon, 1996.

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Bayardo Jr., R.J., Miranker, D.P.: A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem. In: AAAI'96. (1996)

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Roberto J. Bayardo Jr. and Daniel P. Miranker. A complexity analysis of spacebounded learning algorithms for the constraint satisfaction problem. In AAAI'96, 1996.

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R.J. Bayardo Jr., D.P. Miranker, A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem, in: Proc. AAAI-96, Portland, OR, 1996, pp. 298--304. An explanation is said to be relevant if all the decision constraints in it are still valid in the current search state [3].

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R.J. Bayardo, and D.P. Miranker, "A Complexity Analysis of Space-Bounded Learning Algorithms for the Constraint Satisfaction Problem". Proc. 13th National Conference on Artificial Intelligence (AAAI-96), pp. 558--562, 1996.

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Bayardo, R.J. & Miranker, D.P. 1996. A Complexity Analysis of Space-Bounded Learning Algorithms for the Constraint Satisfaction Problem. In proc. AAAI-96: 13th National Conf. on Artificial Intelligence, Portland, Oregon, 298-304.

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