## Tractable structures for constraint satisfaction problems (2006)

Venue: | Handbook of Constraint Programming, part I, chapter 7 |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Dechter06tractablestructures,

author = {Rina Dechter},

title = {Tractable structures for constraint satisfaction problems},

booktitle = {Handbook of Constraint Programming, part I, chapter 7},

year = {2006},

pages = {209--244},

publisher = {Elsevier}

}

### OpenURL

### Abstract

### Citations

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Local computations with probabilities on graphical structures and their application to expert systems (with discussion
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Citation Context ...at memory of O(k i ) is feasible. 5.2 Bibliographical notes Join-tree clustering was introduced in constraint processing by Dechter and Pearl [16] and in probabilistic networks by Spigelhalter et. al =-=[34]-=-. Both methods are based on the characteriza32stion by relational-database researchers that acyclic-databases have an underlying tree-structure, called join-tree, that allows polynomial query processi... |

1052 |
Consistency in Networks of Relations
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Citation Context ...e-based tractability in Inference Almost all the known structure-based techniques rely on the observation that binary constraint networks whose constraint graph is a tree can be solved in linear time =-=[23, 36, 15]-=- in the number of variables. The solution of tree-structured networks are discussed next, and later it is shown how they can be used to facilitate the solution of a general constraint network. 2.1 Sol... |

1016 | Temporal Constraint Networks
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(Show Context)
Citation Context ...atch in that projection. If tuples are ordered lexicographically, which requires O(l · logl) steps, the join operator has a complexity of O(l), yielding an overall complexity of O(r · l · logl) steps =-=[10]-=-. For a more recent analysis see [29]. In summary, THEOREM 2 [10] [correctness and complexity] Algorithm acyclic-solving decides the consistency of an acyclic constraint network, and its complexity is... |

610 |
Constraint Processing
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Citation Context ...etter utilize the benefit of each scheme and allow improved performance guarantees, reduced space complexity and improved average performance. ∗ This chapter is based in parts on chapters 9 and 10 of =-=[12]-=- and on [32] 1sThis chapter focuses on structure-driven constraint processing algorithms. We will start with inference algorithms and show that their performance is controlled by graph parameters such... |

460 |
Theory of Relational Databases
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Citation Context ...r of tuples in each constraint relation . ✷ Several efficient procedures for identifying acyclic networks and for finding a representative join-tree were developed in the area of relational databases =-=[38]-=-. One scheme that proved particularly useful is based on the observation that a network is acyclic if, and only if, its primal graph is both chordal and conformal [5]. A graph is chordal if every cycl... |

443 | Partial constraint satisfaction
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Citation Context ...graph-parameter. In [15, 16] the connection between induced-width and tree-width was recognized through the work of [2] on tree-width and k-trees and partial k-trees, which was made explicit later in =-=[22]-=-. The similarity between variable-elimination and tree-clustering from the constraint perspective was analyzed [16]. Independently of this investigation, the tree-width parameter was undergoing intens... |

401 |
Network-based heuristics for constraint satisfaction problems
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Citation Context ...e-based tractability in Inference Almost all the known structure-based techniques rely on the observation that binary constraint networks whose constraint graph is a tree can be solved in linear time =-=[23, 36, 15]-=- in the number of variables. The solution of tree-structured networks are discussed next, and later it is shown how they can be used to facilitate the solution of a general constraint network. 2.1 Sol... |

358 |
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Citation Context ...t tree-decompositions having a bounded separators’ size in polynomial time, this is a feasible task when the separators are singletons. To this end, we use the graph notion of nonseparable components =-=[17]-=-. DEFINITION 13 (non-separable components) A connected graph G = (V, E) is said to have a separation node v if there exist nodes a and b such that all paths connecting a and b pass through v. A graph ... |

340 |
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Citation Context ...ph-community. It characterizes the best embedding of a graph or a hypergraph in a hypertree. Various connections between hypertrees, chordal graphs and k-trees were made by Arnborg and his colleagues =-=[2, 41]-=-. They showed that finding the smallest tree-width of a graph is NP-complete, but deciding if the graph has a tree-width below a certain constant k is polynomial in k. A recent analysis shows that thi... |

287 |
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Citation Context ...o find a consistent solution having maximum cost. 1.2 Graphical representations Graphical properties of constraint networks were initially investigated through the class of binary constraint networks =-=[23]-=-. A binary constraint network is one in which every constraint scope involves at most two variables. In this case the network can be associated with a constraint graph, where each node represents a va... |

284 |
Enhancement schemes for constraint processing: Backjumping, learning, and cutset decomposition
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Citation Context ...orithms [27, 3] are backtracking search schemes applied to the regular OR space, which uses the problem structure to jump back from a dead-end as far back as possible. In graphbased backjumping (GBJ) =-=[9]-=- each variable maintains a graph-based induced ancestor set which ensures that no solutions are missed when jumping back to its deepest variable. Graph-based backjumping extracts knowledge about depen... |

276 |
Freuder, “The Complexity of Some Polynomial Network Consistency Algorithms for Satisfaction Problems
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Citation Context ... 4. Revise ((Xp(i)), Xi); 5. if the domain of Xp(i) is empty, exit (no solution exists). 6. endfor x 4s¡ ¡ ¡ ¡sFigure 4: Tree-solving algorithm x x x 1 2 3 x 5 Figure 5: A tree network x6 x THEOREM 1 =-=[37]-=- A binary tree constraint problem can be solved in O(nk 2 ) when n is the number of variables and k bounds the domain size. 2.2 Acyclic Networks The notion of constraint trees can be extended beyond b... |

274 | Bucket elimination: A unifying framework for reasoning
- Dechter
- 1999
(Show Context)
Citation Context ...ons to all but the current bucket’s variable. Adaptive-consistency is an instance of a general class of variable elimination algorithms called bucket-elimination that are applicable across many tasks =-=[11]-=-. The complexity of adaptive-consistency is linear in the number of buckets and in the time to process each bucket. Since processing a bucket amounts to solving a constraint subproblem (the computatio... |

263 |
Tree clustering for constraint networks
- Dechter, Pearl
- 1989
(Show Context)
Citation Context ... [2] on tree-width and k-trees and partial k-trees, which was made explicit later in [22]. The similarity between variable-elimination and tree-clustering from the constraint perspective was analyzed =-=[16]-=-. Independently of this investigation, the tree-width parameter was undergoing intensive investigation in the theoretic-graph-community. It characterizes the best embedding of a graph or a hypergraph ... |

251 |
Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs
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(Show Context)
Citation Context ... called the maximal cardinality ordering, (m-ordering); it always assigns the next number to the node having the largest set of already numbered neighbors (breaking ties arbitrarily). It can be shown =-=[46]-=- that in an m-ordered chordal graph, the parent-set of each node, namely,its earlier neighbors in the ordered graph, must be completely connected. If, in addition, the maximal cliques coincide with th... |

207 |
Efficient algorithms for combinatorial problems on graphs with bounded decomposability - a survey
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Citation Context ... (ABC),(AEF ), (CDE) and (ACE). The constraints themselves are symbolically given by the inequalities: A + B ≤ C, A + E ≤ F , C + D ≤ E, A + C ≤ E, where the domains of each variable are the integers =-=[2, 3, 4, 5, 6]-=-. 4 ACsThe dual constraint graph can be viewed as a transformation of a nonbinary network into a special type of binary network: the domain of the c-variables ranges over all possible value combinatio... |

183 |
Nonserial Dynamic Programming
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(Show Context)
Citation Context ... (ABC),(AEF ), (CDE) and (ACE). The constraints themselves are symbolically given by the inequalities: A + B ≤ C, A + E ≤ F , C + D ≤ E, A + C ≤ E, where the domains of each variable are the integers =-=[2, 3, 4, 5, 6]-=-. 4 ACsThe dual constraint graph can be viewed as a transformation of a nonbinary network into a special type of binary network: the domain of the c-variables ranges over all possible value combinatio... |

171 |
A sufficient condition for backtrack–bounded search
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Citation Context ..., DF} 2 1(F) 1 2(F) 3 2(E) 2 4(C) 2 3(E) { RE, H , I E,H,I { RA, D, RA, B, RD, C , DB, C} Figure 25: A graph and its decomposition into non-separable components. THEOREM 15 (non-separable components) =-=[24]-=- If R = (X, D, C), |X| = n, is a constraint network whose constraint graph has non-separable components of at most size r, then the supercluster-tree elimination algorithm, whose buckets are the non-s... |

140 | Treewidth: Algorithmic techniques and results
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(Show Context)
Citation Context ... the graph has a tree-width below a certain constant k is polynomial in k. A recent analysis shows that this task can be accomplished in O(n · f(k)) where f(k) is a very bad exponential function of k =-=[8]-=-. The decomposition into hinges was presented in [35]. As noted any hinge-decomposition is closely related to bi-component tree decomposition of the dual graph whose redundant arcs are removed [30]. T... |

103 | AND/OR search spaces for graphical models
- Dechter, Mateescu
(Show Context)
Citation Context ...subgraph is only solved once and the results are cached and indexed by the context. Searching the AND/OR graphs rather than the AND/OR tree is related to recording no-goods during backtracking search =-=[13]-=-. It can be shown that, THEOREM 17 [14] Given G, a pseudo-tree T of G and its induced width w ∗ the size of the minimal AND/OR search graph based on T is O(n · kw∗), when k bounds the domain size and ... |

86 | Decomposing constraint satisfaction problems using database techniques
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Citation Context ...alled minimal join-graph) and then generate a tree of non-separable components. This idea is very related to another tree-decomposition principle proposed in the literature called hinge-decomposition =-=[35]-=-. Indeed a best hinge decomposition can be obtained in polynomial time, yielding smallest component in a bi-component tree decomposition of the dual graph whose some redundant arcs are removed. For a ... |

80 | Approximation algorithms for the loop cutset problem
- Becker, Geiger
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Citation Context ...-cutset is hard, but various greedy heuristic algorithms were investigated empirically. Several greedy and approximation algorithms for the special case of cycle-cutset can be found in the literature =-=[1]-=-. The general task of finding a minimal w-cutset was addressed in recent papers [21, 7] both for the cutset-decomposition version and for the alternating version. Note that verifying that a given subs... |

80 | A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem
- Bayardo, Miranker
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(Show Context)
Citation Context ... (ABC),(AEF ), (CDE) and (ACE). The constraints themselves are symbolically given by the inequalities: A + B ≤ C, A + E ≤ F , C + D ≤ E, A + C ≤ E, where the domains of each variable are the integers =-=[2, 3, 4, 5, 6]-=-. The dual constraint graph can be viewed as a transformation of a nonbinary network into a special type of binary network: the domain of the c-variables ranges over all possible value combinations pe... |

74 |
Exact genetic linkage computations for general pedigrees
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(Show Context)
Citation Context ...orithm alternates between conditioning and elimination. This scheme was used both for solving SAT problems and for optimization tasks [40, 33] and is currently used for Bayesian networks applications =-=[20, 19]-=-. Clearly, a cutset uncov25sered via the alternating algorithm is also a w-cutset and therefore can be used within the cutsetdecomposition scheme. Both cutset-decomposition and the alternating cutset-... |

71 | A sufficiently fast algorithm for finding close to optimal junction trees
- Becker, Geiger
- 1996
(Show Context)
Citation Context ... (ABC),(AEF ), (CDE) and (ACE). The constraints themselves are symbolically given by the inequalities: A + B ≤ C, A + E ≤ F , C + D ≤ E, A + C ≤ E, where the domains of each variable are the integers =-=[2, 3, 4, 5, 6]-=-. 4 ACsThe dual constraint graph can be viewed as a transformation of a nonbinary network into a special type of binary network: the domain of the c-variables ranges over all possible value combinatio... |

56 | A practical algorithm for finding optimal triangulations - Shoikhet, Geiger - 1997 |

47 | Approximating treewidth, pathwidth, and minimum elimination tree height
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Citation Context ...he OR minimal search graph is bounded exponentially by its pathwidth. The path-width, pw ∗ , of a graph is the minimum tree-width over all tree-decompositions whose trees are chains. It is well known =-=[31]-=- that for any graph w ∗ ≤ pw ∗ ≤ w ∗ · log n. It is also easy to place m ∗ (the minimal pseudo-tree depth) yielding w ∗ ≤ m ∗ ≤ pw ∗ ≤ w ∗ · log n. 5 Summary and Bibliographical Notes 5.1 Structure-ba... |

39 |
A new method for solving constraint satisfaction problems
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Citation Context ...ng any knowledge-base into an acyclic one is exponential in the cluster size, which is characterized by the induced-width or tree-width. At the same time, variable-elimination algorithms developed in =-=[6, 43]-=- and [15] (e.g., adaptive-consistency and bucket-elimination) were also observed to be governed by the same complexity graph-parameter. In [15, 16] the connection between induced-width and tree-width ... |

34 | Maximum likelihood haplotyping for general pedigrees
- Fishelson, Dovgolevsky, et al.
(Show Context)
Citation Context ...orithm alternates between conditioning and elimination. This scheme was used both for solving SAT problems and for optimization tasks [40, 33] and is currently used for Bayesian networks applications =-=[20, 19]-=-. Clearly, a cutset uncov25sered via the alternating algorithm is also a w-cutset and therefore can be used within the cutsetdecomposition scheme. Both cutset-decomposition and the alternating cutset-... |

24 |
Resolution vs. search; two strategies for sat
- Rish, Dechter
(Show Context)
Citation Context ... induced-width higher than w, it will be conditioned upon. The algorithm alternates between conditioning and elimination. This scheme was used both for solving SAT problems and for optimization tasks =-=[40, 33]-=- and is currently used for Bayesian networks applications [20, 19]. Clearly, a cutset uncov25sered via the alternating algorithm is also a w-cutset and therefore can be used within the cutsetdecomposi... |

21 |
Binary join tree
- Shenoy
- 1997
(Show Context)
Citation Context ...elds complexity of O(deg · r · k w∗ +1 ). It is possible to have an implementation of the algorithm whose time complexity will not depend on deg, but this improvement will be more expensive in memory =-=[44, 32]-=-. ✷ Join-tree clustering as tree-decomposition. Algorithm JTC is a specific algorithm for creating the tree-decomposition. Because it generates the full set of solutions for each node, its space compl... |

20 | A.: Unifying tree decompositions for reasoning in graphical models
- Kask, Dechter, et al.
- 2005
(Show Context)
Citation Context ...e the benefit of each scheme and allow improved performance guarantees, reduced space complexity and improved average performance. ∗ This chapter is based in parts on chapters 9 and 10 of [12] and on =-=[32]-=- 1sThis chapter focuses on structure-driven constraint processing algorithms. We will start with inference algorithms and show that their performance is controlled by graph parameters such as tree-wid... |

18 | The impact of AND/OR search spaces on constraint satisfaction and counting
- Dechter, Mateescu
(Show Context)
Citation Context ...sults are cached and indexed by the context. Searching the AND/OR graphs rather than the AND/OR tree is related to recording no-goods during backtracking search [13]. It can be shown that, THEOREM 17 =-=[14]-=- Given G, a pseudo-tree T of G and its induced width w ∗ the size of the minimal AND/OR search graph based on T is O(n · kw∗), when k bounds the domain size and n is the number of variables. We can sh... |

17 |
Performance measurement and analysis of search algorithms
- GASCHNIG
- 1979
(Show Context)
Citation Context ... O(n · k m ) where m is the depth of its backbone pseudo-tree. Moreover, a constraint network that has a tree-width w ∗ has an AND/OR search tree whose size is O(k w∗ ·log n ). Backjumping algorithms =-=[27, 3]-=- are backtracking search schemes applied to the regular OR space, which uses the problem structure to jump back from a dead-end as far back as possible. In graphbased backjumping (GBJ) [9] each variab... |

11 |
The use of lineal spanning trees to represent constraint satisfaction problems
- Freuder, Quinn
- 1987
(Show Context)
Citation Context ...FS tree T with a highlighted solution subtrees are given in Fig. 28. The construction of AND/OR search trees can be guided not just DFS spanning trees but also by pseudo-trees which include DFS trees =-=[25, 26, 3]-=-. Pseudo-trees have the property that every arc of the constraint graph is a back-arc in the pseudo-tree (i.e. it doesn’t connect across different branches). Clearly, any DFS tree and any chain are ps... |

9 | Weighted Hypertree Decompositions and Optimal Query Plans
- Scarcello, Greco, et al.
- 2004
(Show Context)
Citation Context ...tree decomposition having the smallest hypertree width, hw ∗ s. A related problem of finding a tree-decomposition with a bounded tree-width w having the smallest separator, was shown to be polynomial =-=[18]-=-. Finding w ∗ s however, is hard but it is easy for the special case of s = 1 as we show next. 3.2.1 Decomposition into non-separable Ccomponents A special tree-decomposition occurs when all the separ... |

8 |
Approximation algorithms for graphical models
- Kask
(Show Context)
Citation Context ...lds complexity of O(deg · r · exp(w ∗ )). It is possible to have an implementation of the algorithm whose time complexity will not depend on deg, but this improvement will be more expensive in memory =-=[31, 21]-=-. ✷ Join-tree clustering as tree-decomposition. Algorithm JTC is a specific algorithm for creating the tree-decomposition. Because it generates the full set of solutions for each node, its space compl... |

5 |
On the desirability of acyclic database ochemes
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- 1983
(Show Context)
Citation Context |

5 |
Dynamic combination of search and variable-elimination in csp and max-csp
- Larrosa, Dechter
- 2001
(Show Context)
Citation Context ... induced-width higher than w, it will be conditioned upon. The algorithm alternates between conditioning and elimination. This scheme was used both for solving SAT problems and for optimization tasks =-=[40, 33]-=- and is currently used for Bayesian networks applications [20, 19]. Clearly, a cutset uncov25sered via the alternating algorithm is also a w-cutset and therefore can be used within the cutsetdecomposi... |

4 |
On finding w-cutset in bayesian networks
- Bidyuk, Dechter
- 2004
(Show Context)
Citation Context ...lly. Several greedy and approximation algorithms for the special case of cycle-cutset can be found in the literature [1]. The general task of finding a minimal w-cutset was addressed in recent papers =-=[21, 7]-=- both for the cutset-decomposition version and for the alternating version. Note that verifying that a given subset of nodes is a w-cutset can be accomplished in polynomial time (linear in the number ... |

4 |
Optimizing exact genetic linkage computations. RECOMB
- Fishelson, Geiger
- 2003
(Show Context)
Citation Context ...lly. Several greedy and approximation algorithms for the special case of cycle-cutset can be found in the literature [1]. The general task of finding a minimal w-cutset was addressed in recent papers =-=[21, 7]-=- both for the cutset-decomposition version and for the alternating version. Note that verifying that a given subset of nodes is a w-cutset can be accomplished in polynomial time (linear in the number ... |

4 | Bucket elimination and hypertree decompositions. Implementation report - McMahan - 2004 |

3 |
F.: Non-Binary Constraints and Optimal Dual-Graph Representations
- Greco, Scarcello
- 2003
(Show Context)
Citation Context ...inge decomposition can be obtained in polynomial time, yielding smallest component in a bi-component tree decomposition of the dual graph whose some redundant arcs are removed. For a formal proof see =-=[30]-=-. 4 Structure-based tractability in search Search algorithms typically traverse the problem’s space whose paths represent a partial or full solutions. Their main virtue is that they can operate using ... |

3 |
Georg Gottlob and Francesco Scarello. A comparison of structural csp decomposition methods
- Leone
- 1999
(Show Context)
Citation Context ... · logt) because there are m arc in the hyper-tree decomposition, each has at most O(t hw ) tuples so acyclic-solving is O(mt hw log(t hw ) which yields the desired bound. We can summarize, THEOREM 8 =-=[16]-=- Let m be the number of hyper-edges in the hyper-tree decomposition of a constraint network R, hw be its hyper-width and t be a bound on the relation size. A hyper-tree decomposition of a constraint p... |

1 |
Taking adventage of stable sets of variables in constraint staisfaction problems
- Freuder, Quinn
- 1985
(Show Context)
Citation Context ...FS tree T with a highlighted solution subtrees are given in Fig. 28. The construction of AND/OR search trees can be guided not just DFS spanning trees but also by pseudo-trees which include DFS trees =-=[25, 26, 3]-=-. Pseudo-trees have the property that every arc of the constraint graph is a back-arc in the pseudo-tree (i.e. it doesn’t connect across different branches). Clearly, any DFS tree and any chain are ps... |

1 |
Georg Gottlob and Francesco Scarcello. A comparison of structural csp decomposition methods
- Leone
- 2000
(Show Context)
Citation Context ...rity would have a high tree-width even though it can be processed in quadratic time in the input. A different graph parameter that is more sensitive to the hypergraph structure is the hypertree width =-=[28]-=-. It relies on a notion of hypertree decompositions for Constraint Satisfaction and it provides a stronger indicator of tractability than the tree-width. DEFINITION 9 (hypertree decomposition) [28] A ... |

1 | Hypertree decomposition via branch-decomposition - Samer - 2005 |