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## An Optimal Coarse-grained Arc Consistency Algorithm

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Venue: | Artificial Intelligence |

Citations: | 92 - 16 self |

### Citations

1150 |
Consistency in networks of relations
- Mackworth
- 1977
(Show Context)
Citation Context ...lue to each variable such that all constraints in the network are satisfied. For simplicity, in the above definition we consider only binary constraints, omitting the unary constraint on any variable =-=[Mac77a]-=-. Without loss of generality we assume there is only one constraint between each pair of variables. Throughout this paper, n denotes the number of variables, d the size of the largest domain, and e th... |

413 |
Network-based heuristics for constraint satisfaction problems.
- Dechter, Pearl
- 1987
(Show Context)
Citation Context ...tative of a stateof-the-art algorithm because of its good runtime performance over the problems of concern. In addition, a new artificial problem, DOMINO, in the same vein as the problem in Fig. 5 in =-=[DP88]-=-, is designed to study the worst case performance of AC-3. 7 X2Xn−1 Xn X3X1Xn−2 Figure 4: The domino problem Randomly generated problems. For the random instances, we used a model B generator [Pro96].... |

295 |
The Complexity of Some Polynomial Network Consistency Algorithms for Constraint Satisfaction Problems.
- Mackworth, Freuder
- 1985
(Show Context)
Citation Context ... present a new algorithm, AC2001/3.1, which is the first worst case optimal coarse-grained arc consistency algorithm. This result is somewhat surprising since due to the non-optimality result of AC-3 =-=[MF85]-=- from 1985, it is widely held that only fine-grained algorithms have worst case optimality. AC2001/3.1preserves the simplicity of AC-3 while improving on AC-3 in efficiency both in terms of constraint... |

240 |
Arc and path consistency revisited
- Mohr, Henderson
- 1986
(Show Context)
Citation Context ...exity of AC-3 isO(ed3), where e is the number of constraints and d is the size of the maximum domain in a problem. Fine grained algorithms on the other hand enjoy optimal worst case complexity O(ed2) =-=[MH86]-=-. The fine-grained algorithms are also more efficient when applied to networks where much propagation occurs [BFR95, BFR99] while AC-3 is better when there is very little propagation. In this paper, w... |

214 | A Generic Arc Consistency Algorithm and Its Specializations.
- Hentenryck, Deville, et al.
- 1992
(Show Context)
Citation Context ...e and value based constraint graphs can lead to worst case optimal algorithms, we consider their strength on some special constraints: functional, monotonic and anti-functional. For more details, see =-=[VDT92]-=- and [ZY00]. Coarse grained algorithms can be easily adapted to process monotonic and anti-monotonic constraints in a time complexity of O(ed) (e.g., using AC2001/3.1). Fine grained algorithms 4The on... |

152 | Arc-consistency and arc-consistency again.
- Bessiere
- 1994
(Show Context)
Citation Context ...e time and space complexities. So, an interesting question here is “What are the differences between AC2001/3.1 and AC-6 in terms of constraint checks?”. Let us first briefly recall the AC-6 behavior =-=[Bes94]-=-. AC-6 looks for one support (the first one or smallest one with respect to the ordering <d) for each value (xi; a) with respect to each constraint cij to prove that a is currently viable. When (xj ; ... |

135 | Arcconsistency for general constraint networks: Preliminary results
- Bessiere, Ch
- 1997
(Show Context)
Citation Context ...(ed2), and AC2001/3.1, AC-6 and AC-7 have O(ed). When dealing with non-binary constraints, GAC3 [Mac77b] has a O(er3dr+1) time complexity, GAC2001/3.1 is in O(er2dr), while GAC4 [MM88] and GAC-schema =-=[BR97]-=- are in O(erdr). GAC4 is a factor r better than GAC2001/3.1 because it computes the dr possible constraint checks on a constraint once and for all at the beginning, storing the information in lists of... |

134 |
An empirical study of phase transitions in Binary Constraint Satisfaction Problems.
- Prosser
- 1996
(Show Context)
Citation Context ...n [DP88], is designed to study the worst case performance of AC-3. 7 X2Xn−1 Xn X3X1Xn−2 Figure 4: The domino problem Randomly generated problems. For the random instances, we used a model B generator =-=[Pro96]-=-. The parameters are hN; D; C; T i, where N is the number of variables, D the size of the domains, C the number of constraints (the density p1 is equal to 2C=N (N 1)), and T the number of forbidde... |

111 |
Good old discrete relaxation.
- Mohr, Masini
- 1988
(Show Context)
Citation Context ... has a complexity of O(ed2), and AC2001/3.1, AC-6 and AC-7 have O(ed). When dealing with non-binary constraints, GAC3 [Mac77b] has a O(er3dr+1) time complexity, GAC2001/3.1 is in O(er2dr), while GAC4 =-=[MM88]-=- and GAC-schema [BR97] are in O(erdr). GAC4 is a factor r better than GAC2001/3.1 because it computes the dr possible constraint checks on a constraint once and for all at the beginning, storing the i... |

100 | Refining the Basic Constraint Propagation Algorithm’. In: - Bessiere, Regin - 2001 |

74 | Experimental case studies of backtrack vs. waltz-type vs. new algorithms for satisficing assignment problems - Gaschnig - 1978 |

70 |
On reading sketch maps,”
- Mackworth
- 1977
(Show Context)
Citation Context ...ast((xi; a); cj) else delete a from Di DELETE true return DELETE end Figure 7: Procedure REVISE for GAC2001/3.1 14 In Fig. 8, we present a version of the main algorithm based on the one proposed in =-=[Mac77b]-=-. It is a brute force propagation schema that does not take into account the fact that in practice some of the constraints may have ad hoc propagators. Thus the algorithm may have to be adapted depend... |

70 | Radio link frequency assignment - Cabon, Givry, et al. - 1999 |

65 | Using inference to reduce arc consistency computation - Bessière, Freuder, et al. - 1995 |

64 |
Relational consistency algorithms and their application in finding subgraph and graph isomorphisms
- McGregor
- 1979
(Show Context)
Citation Context ... all relevant arcs. When we dequeue a variable from the queue, we just revise all constraints pointing to this variable. The method is also called variable oriented propagation. This idea appeared in =-=[McG79]-=- and in [CJ98]. In this method, for each variable, one more constraint is revised than in the original algorithm AC-3. However, it seems that the savings from enqueue operations well compensates this ... |

61 | Using constraint metaknowledge to reduce arc consistency computation
- Bessière, Freuder, et al.
- 1999
(Show Context)
Citation Context ...ompetitive with these fine-grained algorithms. There have been many experimental studies on the performance of general arc consistency algorithms [Wal93, Bes94, BFR99]. Here, we take problems used in =-=[BFR99]-=-, namely some random CSPs and Radio Link Frequency Assignment Problems (RLFAPs). Given the experimental results of [BFR99], AC-6 is chosen as a representative of a stateof-the-art algorithm because of... |

51 | Ordering heuristics for arc consistency algorithms
- Wallace, Freuder
- 1992
(Show Context)
Citation Context ...r computation model. For example, to save the number of constraint checks, the bidirectionality can be employed also in coarse-grained algorithm, e.g., in [Gas78, LBH03]. Other propagation heuristics =-=[WF92]-=- such as propagating deletion first [BFR99] are also applicable to the algorithms of both models. We have delineated the AC algorithms which shows that AC2001/3.1 and AC-6 are methodologically differe... |

49 | The constrainedness of arc consistency - Gent, MacIntyre, et al. - 1997 |

46 | Making AC-3 an Optimal Algorithm’. In: - Zhang, Yap - 2001 |

28 |
Arc consistency for factorable relations
- Perlin
- 1992
(Show Context)
Citation Context ...the total number of operations during the execution of a fine-grained algorithm will be of the same order as the number of edges in the value based constraint graph: O(ed2). As far as we know, Perlin =-=[Per92]-=- is the first to make value based constraint graph explicit in arc consistency enforcing algorithm. Given a computation model of propagation, the algorithms differ in the implementation details. Under... |

23 | F.: Exploiting multidirectionality in coarse-grained arc consistency algorithms - Lecoutre, Boussemart, et al. |

20 |
P.: Efficient path-consistency propagation
- Chmeiss, Jegou
- 1998
(Show Context)
Citation Context ...rcs. When we dequeue a variable from the queue, we just revise all constraints pointing to this variable. The method is also called variable oriented propagation. This idea appeared in [McG79] and in =-=[CJ98]-=-. In this method, for each variable, one more constraint is revised than in the original algorithm AC-3. However, it seems that the savings from enqueue operations well compensates this cost in our ex... |

16 | Path consistency revisited.
- Singh
- 1995
(Show Context)
Citation Context ...s of possible choices for i; j; k; a; b, that isO(n3d2). 2 The PC2001/3.1 has time complexity of O(n3d3) and space complexity of O(n3d2) which is the same bounds as the best known results obtained in =-=[Sin96]-=-. The algorithm in [Sin96] employs a supported list for each value of a variable and propagates the removal of values in a fashion of AC-6. Compared with the supported list, the structure 13 Last((i; ... |

14 |
Arc consistency on n-ary monotonic and linear constraints
- ZHANG, YAP
- 2000
(Show Context)
Citation Context ...based constraint graphs can lead to worst case optimal algorithms, we consider their strength on some special constraints: functional, monotonic and anti-functional. For more details, see [VDT92] and =-=[ZY00]-=-. Coarse grained algorithms can be easily adapted to process monotonic and anti-monotonic constraints in a time complexity of O(ed) (e.g., using AC2001/3.1). Fine grained algorithms 4The only value co... |

11 |
Random uniform csp generators
- Frost, Bessiere, et al.
- 1996
(Show Context)
Citation Context ...s, C the number of constraints (the density p1 is equal to 2C=N (N 1)), and T the number of forbidden tuples per constraint (the tightness p2 is equal to T=D2). We used the generator available in =-=[FBDR96]-=-. For each class of problems tested, we ran the first 50 instances generated using the initial seed 1964 (as in [BFR99]). RLFAP. The radio link frequency assignment problem (RLFAP) is to assign freque... |

4 |
AC-3 is almost always better than AC-4 for establishing arc consistency in CSPs
- Why
- 1993
(Show Context)
Citation Context ...mining in the context of coarse-grained algorithms those techniques (e.g., bidirectionality and other heuristics or meta knowledge) mainly employed in fine-grained algorithms. Detailed experiments in =-=[Wal93]-=- show the advantage of AC-3 over AC-4. Our work complements this by providing a way to make coarse-grained algorithms to be worst case optimal. 7 Conclusion This paper presents AC2001/3.1, a coarse-gr... |

3 |
Contradicting conventional wisdosdom in constraint satisfaction
- Sabin, Freuder
- 1994
(Show Context)
Citation Context ....2 Maintaining Arc Consistency during Search The second set of experiments we present in this section shows the behaviour of AC2001/3.1 when arc consistency is maintained during search (MAC algorithm =-=[SF94]-=-) to find the first solution. We present results for all the instances contained in the FullRLFAP archive for which more than 2 seconds were needed to find a solution or to prove that none exists. It ... |

2 | User’s manual - Laburthe - 2000 |