### Citations

1138 |
Consistency in networks of relations
- Mackworth
- 1977
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Citation Context ...ints, tractability 1 Introduction The constraint satisfaction problem provides a natural framework for expressing many combinatorial problems in computer science. Since the general problem is NP-hard =-=[15]-=-, an important question is how to restrict the problem to ensure tractability. This research has mainly followed two dierent paths: restricting the scope of the constraints [10, 11], i.e. which varia... |

193 | Reasoning about temporal relations: a maximal tractable subset of Allen’s interval algebra
- Nebel, Bürckert
- 1995
(Show Context)
Citation Context ...be constructed from existing classes. Several important classes of tractable constraints can be obtained by their method such as the Horn and Krom fragments of propositional logic, the ORD-Horn class =-=[16]-=- and the classes of max-closed and connected row-convex constraints [13, 9]. The investigation of disjunctive constraints was continued in Broxvall & Jonsson [5] where all tractable disjunctive classe... |

174 |
A sufficient condition for backtrack-bounded search
- Freuder
- 1985
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Citation Context ...al problem is NP-hard [15], an important question is how to restrict the problem to ensure tractability. This research has mainly followed two dierent paths: restricting the scope of the constraints =-=[10, 11]-=-, i.e. which variables may be constrained with other variables, or restricting the constraints [9, 13, 20], i.e. the allowed values for mutually constrained variables. In this paper, we will only cons... |

174 | A comparison of structural CSP decomposition methods
- Gottlob, Leone, et al.
- 1999
(Show Context)
Citation Context ...al problem is NP-hard [15], an important question is how to restrict the problem to ensure tractability. This research has mainly followed two dierent paths: restricting the scope of the constraints =-=[10, 11]-=-, i.e. which variables may be constrained with other variables, or restricting the constraints [9, 13, 20], i.e. the allowed values for mutually constrained variables. In this paper, we will only cons... |

141 | On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the region connection calculus
- Renz, Nebel
- 1999
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Citation Context ...consistency for ? A , ? B , ? C and D . Proof: Let ? 0 = f S r2R (r) j R 2 ? A g (where is dened as in Lemma 22) and note that ? 0 R 28 5 [14]. Since R 28 5 can be decided by path-consistency =-=[19]-=-, Lemma 22 implies that path-consistency decides ? A . Similarly, we can verify that path-consistency decides ? B ; in this case, ? 00 = f S r2R (r) j R 2 ? B g R 14 5 [14]. Showing that R 14 5 is ... |

109 | Spatial reasoning with propositional logics
- Bennett
- 1994
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Citation Context ... begin by proving a useful connection (Lemma 22) between RCC-5 and the point algebra for partially ordered time which in turn will be needed to prove that path-consistency decides ? A and ? B . RCC-5 =-=[2]-=- is based on the notions of regions and binary relations on them. A region p is a regular open set of a topological space. Regions themselves do not have to be internally connected, i.e. a region may ... |

47 | Maximal tractable fragments of the Region Connection Calculus: A complete analysis
- Renz
- 1999
(Show Context)
Citation Context ... by checking path-consistency, we present a fairly simple method for verifying the 1independence property. This method builds on a somewhat surprising connection between 1-independence and renements =-=[17]-=-. Loosely speaking, a renement is a way of reducing one CSP problem to another and it has the property that if the second problem can be decided by path-consistency, then path-consistency decides the... |

29 | Hentenryck, Constraint satisfaction over connected row convex constraints
- Deville, Barette, et al.
- 1997
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Citation Context ...roof: If path-consistency computes minimal labels, then it makes :R explicit for every R 2 . 2 Examples of when path-consistency computes minimal labels can, for instance, be found in Deville et al. =-=[9]-=- and Bessiere et al. [3]. 4.2 Computational experience We will demonstrate that many 1-independence results can be obtained by using the renement method. We shall show that all independence results ... |

24 | Building tractable disjunctive constraints
- Cohen, Jeavons, et al.
- 2000
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Citation Context ...tematicity in this search, it is of considerable interest to investigate how tractable constraint types may be combined in order to yield more general problems which are still tractable. Cohen et al. =-=[8]-=- have studied so-called `disjunctive constraints', i.e. constraints which have the form of the disjunction of two constraints of specied types. They identied certain properties which allow for new t... |

22 | Expressive Power and Complexity in Algebraic Logic - Hirsch - 1997 |

18 |
On the minimality and decomposability of row-convex constraint networks
- Beek, Dechter
- 1995
(Show Context)
Citation Context .... This research has mainly followed two dierent paths: restricting the scope of the constraints [10, 11], i.e. which variables may be constrained with other variables, or restricting the constraints =-=[9, 13, 20]-=-, i.e. the allowed values for mutually constrained variables. In this paper, we will only consider problems where the constraints are restricted. Quite a large number of tractable subclasses of the CS... |

10 | Global consistency in interval algebra networks: tractable subclasses
- Bessiere, Isli, et al.
- 1996
(Show Context)
Citation Context ... computes minimal labels, then it makes :R explicit for every R 2 . 2 Examples of when path-consistency computes minimal labels can, for instance, be found in Deville et al. [9] and Bessiere et al. =-=[3]-=-. 4.2 Computational experience We will demonstrate that many 1-independence results can be obtained by using the renement method. We shall show that all independence results for the point algebras fo... |

5 |
A complete classi of tractability in RCC-5
- Jonsson, Drakengren
- 1997
(Show Context)
Citation Context ...ed analogously. 2 Theorem 23 Path-consistency decides consistency for ? A , ? B , ? C and D . Proof: Let ? 0 = f S r2R (r) j R 2 ? A g (where is dened as in Lemma 22) and note that ? 0 R 28 5 =-=[14]-=-. Since R 28 5 can be decided by path-consistency [19], Lemma 22 implies that path-consistency decides ? A . Similarly, we can verify that path-consistency decides ? B ; in this case, ? 00 = f S r2R ... |

4 |
Tarjan, A linear time algorithm for testing the truth of certain quanti Boolean formulas
- Aspvall, Plass, et al.
- 1979
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Citation Context ...is tractable) and that the test in line 8 can be performed in polynomial time by using some tractable algorithm for showing the satisability of 2CNF formulae (such as the algorithm by Aspvall et al. =-=[1]-=-). 2 Proof: (of Theorem 12) The if direction follows from Lemma 13. To show the other direction we assume to the contrary that CspSat(?[ 2 ) is tractable but is not 2-independent of ?. 12 This impl... |

4 | Disjunctive temporal reasoning in partially ordered time structures
- Broxvall, Jonsson
- 2000
(Show Context)
Citation Context ...opositional logic, the ORD-Horn class [16] and the classes of max-closed and connected row-convex constraints [13, 9]. The investigation of disjunctive constraints was continued in Broxvall & Jonsson =-=[5]-=- where all tractable disjunctive classes for reasoning about partially and totally ordered time were identied. Somewhat surprisingly, all of these tractable classes can be obtained by using 1-indepen... |

3 | Re and independence: A simple method for identifying tractable disjunctive constraints - Broxvall, Jonsson, et al. - 2000 |

3 |
New tractable classes from old
- Cohen, Jeavons, et al.
- 2000
(Show Context)
Citation Context ...ults so we pose the following problem: Open question 1. Assume CspSat(? [ ) is tractable. What is a necessary and sucient condition for tractability of CspSat(? _). Ideas taken from Cohen et al. =-=[7]-=- can probably be used for answering this question if we restrict ? and to be relations over disjoint domains. We have provided a method for automatically deciding the 1-independence property based o... |

2 |
Towards a complete classi of tractability in point algebras for nonlinear time
- Broxvall, Jonsson
- 1999
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Citation Context ...ther x y nor y x The point algebra for partially ordered time has been throughly investigated earlier and a total classication with respect to tractability has been given in Broxvall and Jonsson =-=[4]-=-. In Broxvall and Jonsson [5] the sets of relations in Table 1 are dened and it is proven that ? A _ A , ? B _ B , ? C _ C and D are the unique maximal tractable disjunctive classe... |

2 | On the complexity of binary relations - Renz |

1 |
Tractable constraints on ordered domains, Arti Intelligence 79
- Jeavons, Cooper
- 1996
(Show Context)
Citation Context .... This research has mainly followed two dierent paths: restricting the scope of the constraints [10, 11], i.e. which variables may be constrained with other variables, or restricting the constraints =-=[9, 13, 20]-=-, i.e. the allowed values for mutually constrained variables. In this paper, we will only consider problems where the constraints are restricted. Quite a large number of tractable subclasses of the CS... |