## The Phase Transition Behaviour of Maintaining Arc Consistency (1995)

Venue: | In Proceedings of ECAI-96 |

Citations: | 24 - 6 self |

### BibTeX

@INPROCEEDINGS{Grant95thephase,

author = {Stuart A. Grant and Barbara M. Smith},

title = {The Phase Transition Behaviour of Maintaining Arc Consistency},

booktitle = {In Proceedings of ECAI-96},

year = {1995},

pages = {175--179}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper, we study two recently presented algorithms employing a "full look-ahead" strategy: MAC (Maintaining Arc Consistency); and the hybrid MAC-CBJ, which combines conflict-directed backjumping capability with MAC. We observe their behaviour with respect to the phase transition properties of randomly-generated binary constraint satisfaction problems, and investigate the benefits of maintaining a higher level of consistency during search by comparing MAC and MAC-CBJ with the FC and FC-CBJ algorithms, which maintain only node consistency. The phase transition behaviour that has been observed for many classes of problem as a control parameter is varied has prompted a flurry of research activity in recent years. Studies of these transitions, from regions where most problems are easy and soluble to regions where most are easy but insoluble, have raised a number of important issues such as the phenomenon of exceptionally hard problems ("ehps") in the easy-soluble region, and the grow...

### Citations

1789 | Random Graphs
- BOLLOBAS
- 2001
(Show Context)
Citation Context ...on the algorithm being used: even a minor change in variable instantiation order may convert an ehp into a much easier problem. 2 Theoretical work on the connectivity of random graphs can be found in =-=[3]-=-. 6 So although individual ehps, and what makes them so difficult for a particular algorithm, should be investigated, we can also ask about ehps in relation to populations of problems and in relation ... |

1000 |
Consistency in networks of relations
- Mackworth
- 1977
(Show Context)
Citation Context ...heir remaining domains, and following Sabin & Freuder [34] we have termed this algorithm MAC (Maintaining Arc Consistency). Our implementations are based on Mackworth's AC-3 arc consistency algorithm =-=[24]-=- whereas [34] reports an implementation based on Mohr & Henderson's AC-4 [26]. As with forward checking, MAC is a chronological backtracker, and so we also investigate the addition of conflict-directe... |

577 | Where the really hard problems are - Cheeseman, Kanefsky, et al. - 1991 |

457 |
G.: Increasing tree-search efficiency for constraint satisfaction problems
- Haralick, Elliot
- 1980
(Show Context)
Citation Context ... become established as the favoured complete methods for searching constraint satisfaction and other classes of hard problems. The most common example of such a technique is the forward checking (FC) =-=[20]-=- algorithm, which maintains node consistency among the future variables by removing values from their domains which are inconsistent with the current partial solution. It has been shown [20, 27, 29] t... |

389 |
Network-based heuristics for constraint-satisfaction problems
- Dechter, Pearl
- 1987
(Show Context)
Citation Context ... performance is assessed. There has been a rapid move away from attempting to classify algorithms on the basis of limited testing on sets of homogenous problems, such as n-queens or the zebra problem =-=[8]-=-, towards attempting to classify algorithms in terms of performance on large samples of problems of varying size, topology and position in relation to the phase transition (for instance the study by T... |

359 | Dynamic backtracking
- Ginsberg
- 1993
(Show Context)
Citation Context ... suggests that all exceptionally hard problems can be eliminated by a search strategy employing a sufficiently intelligent backtracker, and presents experiments using dependency-directed backtracking =-=[18]-=- which records nogoods during search. However, Baker admits that this algorithm `has an increasing [spatial] overhead as problems get harder' which once again may be prohibitively expensive. The secon... |

346 | Hybrid algorithms for the constraint satisfaction problem
- Prosser
- 1993
(Show Context)
Citation Context ...king (FC) [20] algorithm, which maintains node consistency among the future variables by removing values from their domains which are inconsistent with the current partial solution. It has been shown =-=[20, 27, 29]-=- that the additional overheads associated with making more intelligent forward search moves by forward checking are often greatly outweighed by a large reduction in the size of the search tree. 1 The ... |

283 |
and easy distributions of sat problems
- Hard
- 1992
(Show Context)
Citation Context ... region becomes narrower, and is instantaneous in the limit [45, 46, 47]. Phase transition behaviour has been reported in an increasing number of classes of problem, including satisfiability problems =-=[6, 14, 22, 25]-=-, Hamiltonian paths [5], and the travelling salesman problem [16, 17]. Williams and 2 Hogg [45, 46, 47] have developed approximations to the cost of finding the first solution and to the probability t... |

210 |
Arc and path consistency revisited
- Mohr, Henderson
- 1986
(Show Context)
Citation Context ...s algorithm MAC (Maintaining Arc Consistency). Our implementations are based on Mackworth's AC-3 arc consistency algorithm [24] whereas [34] reports an implementation based on Mohr & Henderson's AC-4 =-=[26]-=-. As with forward checking, MAC is a chronological backtracker, and so we also investigate the addition of conflict-directed backjumping (CBJ) [29] to MAC to produce the potentially more efficient MAC... |

203 | E.: Contradicting conventional wisdom in constraint satisfaction
- Sabin, Freuder
- 1994
(Show Context)
Citation Context ... Backtracking) [10]. The algorithm re-establishes arc consistency after every instantiation in the subproblem formed by the future variables and their remaining domains, and following Sabin & Freuder =-=[34]-=- we have termed this algorithm MAC (Maintaining Arc Consistency). Our implementations are based on Mackworth's AC-3 arc consistency algorithm [24] whereas [34] reports an implementation based on Mohr ... |

201 | Experimental results on the cross-over point in satisfiability problems
- Crawford, Auton
- 1993
(Show Context)
Citation Context ... region becomes narrower, and is instantaneous in the limit [45, 46, 47]. Phase transition behaviour has been reported in an increasing number of classes of problem, including satisfiability problems =-=[6, 14, 22, 25]-=-, Hamiltonian paths [5], and the travelling salesman problem [16, 17]. Williams and 2 Hogg [45, 46, 47] have developed approximations to the cost of finding the first solution and to the probability t... |

174 | Performance measurement and analysis of certain search algorithms
- Gaschnig
- 1979
(Show Context)
Citation Context ...vestigation forms the basis of the work presented here. We study a version of the algorithm originally reported by Gaschnig as CS2 [9] and later as DEEB (Domain Element Elimination with Backtracking) =-=[10]-=-. The algorithm re-establishes arc consistency after every instantiation in the subproblem formed by the future variables and their remaining domains, and following Sabin & Freuder [34] we have termed... |

123 | A C++ Implementation of CLP
- Puget
- 1994
(Show Context)
Citation Context ...ed in [7], [13] and [21]. 3 1.2 A history of MAC Although MAC is a popular technique employed by the constraint programming community, and is used by many constraint solving tools such as ILOG Solver =-=[32]-=-, its application by the constraint satisfaction community has until very recently been passed over in favour of the lesser level of look-ahead provided by FC. A likely explanation for this stems from... |

122 |
Constraint satisfaction algorithms
- Nadel
- 1989
(Show Context)
Citation Context ...king (FC) [20] algorithm, which maintains node consistency among the future variables by removing values from their domains which are inconsistent with the current partial solution. It has been shown =-=[20, 27, 29]-=- that the additional overheads associated with making more intelligent forward search moves by forward checking are often greatly outweighed by a large reduction in the size of the search tree. 1 The ... |

113 |
An empirical study of phase transitions in binary constraint satisfaction problems
- Prosser
- 1996
(Show Context)
Citation Context ...he set of h30; 10; p 1 i problem classes studied is listed in Table 1 along with additional background information. The table includes the theoretical critical values of p 2 ,sp 2crit as presented in =-=[28, 38]-=-, at which average search effort is expected to be maximal. This value should also correspond with the crossover point at which 50% of problems are satisfiable, and is calculated as:sp 2crit = 1 \Gamm... |

112 | A theoretical evaluation of selected backtracking algorithms
- Kondrak, Beek
- 1997
(Show Context)
Citation Context ...lues in its domain, until successful or forced to backtrack, each of these trial instantiations is counted as a search tree node visited. This definition of nodes visited corresponds to that given in =-=[23]-=-. This measure is both environment and implementation independent, and so allows direct comparison of any MAC implementation. However, it may not reflect the varying levels of search effort required d... |

89 |
Backtrack programming
- Golomb, Baumert
- 1965
(Show Context)
Citation Context ...wever, subsequent studies [41] have shown that densely-constrained ehps may occur with the most naive algorithms having no form of look-ahead capability, for instance plain chronological backtracking =-=[19]. Having examined wh-=-at might be termed the "macroscopic" behaviour of ehps for this algorithm, we then studied the "microscopic" behaviour of a selection of individual ehps in an attempt to understand... |

88 | The hardest constraint problems: A double phase transition
- Williams, Hogg
- 1994
(Show Context)
Citation Context ... be made. However, recent studies have highlighted a phenomenon that complicates the phase transition model. The existence of exceptionally hard problems ("ehps") has been reported by Hogg &=-= Williams [21]-=- in graph colouring, Gent & Walsh [12] in satisfiability problems, and by Smith [35] and Smith & Grant [40] in CSPs. These studies show that although there is a welldefined peak in the median cost of ... |

78 | Easy problems are sometimes hard
- Gent, Walsh
- 1994
(Show Context)
Citation Context ... highlighted a phenomenon that complicates the phase transition model. The existence of exceptionally hard problems ("ehps") has been reported by Hogg & Williams [21] in graph colouring, Gen=-=t & Walsh [12]-=- in satisfiability problems, and by Smith [35] and Smith & Grant [40] in CSPs. These studies show that although there is a welldefined peak in the median cost of finding a solution in the region of th... |

76 |
Phase transition and the mushy region in constraint satisfaction problems
- Smith
- 1994
(Show Context)
Citation Context ... of problem solubility falls from close to 1 to close to 0, and the median cost of searching these problems is highest, reaching a peak at the crossover point where 50% of problems are soluble. Smith =-=[35, 37]-=- has termed this region the mushy region; as problem size increases, the mushy region becomes narrower, and is instantaneous in the limit [45, 46, 47]. Phase transition behaviour has been reported in ... |

72 | Exploiting the deep structure of constraint problems
- Williams, Hogg
- 1994
(Show Context)
Citation Context ...ssover point where 50% of problems are soluble. Smith [35, 37] has termed this region the mushy region; as problem size increases, the mushy region becomes narrower, and is instantaneous in the limit =-=[45, 46, 47]-=-. Phase transition behaviour has been reported in an increasing number of classes of problem, including satisfiability problems [6, 14, 22, 25], Hamiltonian paths [5], and the travelling salesman prob... |

71 |
Binary Constraint Satisfaction Problems: Some are Harder than Others
- Prosser
- 1994
(Show Context)
Citation Context ...ove insolubility, or may partially prune the domains of a number of the problem variables and conceptually make search easier by reducing the potential search space. However, it has been demonstrated =-=[30, 34]-=- that algorithms employing dynamic variable ordering can occasionally perform more poorly as a result of the pruning effects of constraint propagation. We conducted a brief study of the effects of pre... |

64 | The TSP Phase Transition
- Gent, Walsh
- 1996
(Show Context)
Citation Context ... region becomes narrower, and is instantaneous in the limit [45, 46, 47]. Phase transition behaviour has been reported in an increasing number of classes of problem, including satisfiability problems =-=[6, 14, 22, 25]-=-, Hamiltonian paths [5], and the travelling salesman problem [16, 17]. Williams and 2 Hogg [45, 46, 47] have developed approximations to the cost of finding the first solution and to the probability t... |

42 | Sparse constraint graphs and exceptionally hard problems
- Smith, Grant
- 1995
(Show Context)
Citation Context ...g in the sample at the same constraint tightness, and much more difficult (again by at least an order of magnitude) than 99% of the sample problems in the phase transition. As in our previous studies =-=[36, 39]-=-, we found no ehps in the easy-soluble regions that are insoluble problems: we continue to conjecture that in the case of CSPs such problems must be exceptionally rare 14 , even among ehps. Figure 6 s... |

41 | Using deep structure to locate hard problems
- Williams, Hogg
- 1992
(Show Context)
Citation Context ...ssover point where 50% of problems are soluble. Smith [35, 37] has termed this region the mushy region; as problem size increases, the mushy region becomes narrower, and is instantaneous in the limit =-=[45, 46, 47]-=-. Phase transition behaviour has been reported in an increasing number of classes of problem, including satisfiability problems [6, 14, 22, 25], Hamiltonian paths [5], and the travelling salesman prob... |

40 | Why AC-3 is almost always better than AC-4 for establishing arc consistency in csps
- Wallace
- 1993
(Show Context)
Citation Context ...e problem arc consistent, the queue initially contains every directed arc in the constraint graph. 3 In fact, the improved efficiency of more complex AC algorithms has recently been thrown into doubt =-=[44]-=- 4 Prosser notes that it may be more appropriate to refer to our version of MAC as MAC3, and to Sabin & Freuder's version as MAC4, etc., but within the scope of this paper we will continue to refer si... |

35 | MAC-CBJ: Maintaining Arc Consistency with ConflictDirected Backjumping
- Prosser
- 1995
(Show Context)
Citation Context ...with forward checking, MAC is a chronological backtracker, and so we also investigate the addition of conflict-directed backjumping (CBJ) [29] to MAC to produce the potentially more efficient MAC-CBJ =-=[31]-=-. We study the behaviour of MAC and MAC-CBJ with respect to the phase transition behaviour of randomly-generated sets of binary constraint satisfaction problems (CSPs), enabling us to apply the algori... |

30 |
Critical behaviour in the satisfiability of random boolean expressions
- Kirkpatrick, Selman
- 1994
(Show Context)
Citation Context |

29 | The phase transition in constraint satisfaction problems: A closer look at the mushy region
- Smith
- 1993
(Show Context)
Citation Context ...uring, k-SAT) and for the general case. In the case of binary CSPs, Prosser [30] has carried out an extensive series of experiments investigating the phase transition, and Smith [37] and Smith & Dyer =-=[38]-=- have discussed the extent to which it is possible to predict its location. The importance of the phase transition cannot be overstated. The increasing set of problem classes for which phase transitio... |

27 | Scaling effects in the CSP phase transition
- Gent, MacIntyre, et al.
- 1995
(Show Context)
Citation Context ...t may not reflect the varying levels of search effort required during each trial instantiation, and so is not as analogous to real time as consistency checks. 7 Recent work on scaling effects in CSPs =-=[11]-=- uses a control parameterswhose value is a constant times fl. 8 For example, whether a check should be counted if the variables are unconstrained: we do not do so. 12 CPU time: the amount of cpu time ... |

24 |
A constraint satisfaction method for inference making
- Gaschnig
- 1974
(Show Context)
Citation Context ...ntly become the subject of a large amount of research, and such an investigation forms the basis of the work presented here. We study a version of the algorithm originally reported by Gaschnig as CS2 =-=[9]-=- and later as DEEB (Domain Element Elimination with Backtracking) [10]. The algorithm re-establishes arc consistency after every instantiation in the subproblem formed by the future variables and thei... |

23 | The hardest random SAT problems, in
- Gent, Walsh
- 1994
(Show Context)
Citation Context ...behaviour when comparing algorithm performance. 1 To date, no exceptionally hard soluble problems have been reported in studies of incomplete local search methods, for instance those reported in [7], =-=[13]-=- and [21]. 3 1.2 A history of MAC Although MAC is a popular technique employed by the constraint programming community, and is used by many constraint solving tools such as ILOG Solver [32], its appli... |

18 | An attempt to map the performance of a range of algorithm and heuristic combinations
- Tsang, Borrett, et al.
- 1995
(Show Context)
Citation Context ...empting to classify algorithms in terms of performance on large samples of problems of varying size, topology and position in relation to the phase transition (for instance the study by Tsang, et al. =-=[42]-=-). Knowledge of phase transitions should eventually allow problems to be analysed before any search is undertaken, and accurate predictions about solution probability, likely difficulty, and most suit... |

17 |
Extending Deep Structure
- Williams, Hogg
- 1993
(Show Context)
Citation Context ...ssover point where 50% of problems are soluble. Smith [35, 37] has termed this region the mushy region; as problem size increases, the mushy region becomes narrower, and is instantaneous in the limit =-=[45, 46, 47]-=-. Phase transition behaviour has been reported in an increasing number of classes of problem, including satisfiability problems [6, 14, 22, 25], Hamiltonian paths [5], and the travelling salesman prob... |

12 |
An arc-consistency algorithm optimal in the number of constraint checks
- Bessiere, Regin
- 1994
(Show Context)
Citation Context ...of preprocessing using AC-3 on each of the problem classes listed in Tables 1 and 2. This study is similar in style to that of Borrett & Tsang [4], who reported the effects of preprocessing with AC-6 =-=[2]-=- in terms of the amount of useful work (i.e. the amount of domain pruning) performed on various CSP problem classes. In our study, we look at both the amount of variable domain pruning that occurs and... |

9 | An empirical investigation into the exceptionally hard problems
- Davenport, Tsang
- 1995
(Show Context)
Citation Context ...n used the median rather than the mean as a measure of average difficulty. To date no complete 1 search method has been shown to be completely immune from ehps, although studies of various algorithms =-=[40, 41, 7, 1]-=- have shown that their incidence and magnitude varies greatly between search methods. It is clearly important, therefore, to consider relative ehp behaviour when comparing algorithm performance. 1 To ... |

9 | Where the Exceptionally Hard Problems are
- Smith, Grant
- 1995
(Show Context)
Citation Context ...om CSP problem classes has many other potential uses: for example, the data has been used with that for some other algorithms on the same problem classes to establish an ehp "hierarchy", pre=-=sented in [41]-=-. Acknowledgments Stuart Grant is partly supported by a studentship from British Telecom plc. We are extremely grateful to Patrick Prosser and Ian Gent at the University of Strathclyde for their inval... |

8 | Intelligent backtracking on the hardest constraint problems
- Baker
- 1995
(Show Context)
Citation Context ...n used the median rather than the mean as a measure of average difficulty. To date no complete 1 search method has been shown to be completely immune from ehps, although studies of various algorithms =-=[40, 41, 7, 1]-=- have shown that their incidence and magnitude varies greatly between search methods. It is clearly important, therefore, to consider relative ehp behaviour when comparing algorithm performance. 1 To ... |

8 | Redundant Hidden Variables in Finite Domain Constraint Problems
- Rossi
- 1995
(Show Context)
Citation Context ...ons) is ineffective. If such a constraint gap exists for CSPs, this AC data (concerning propagation of nogoods) together with data concerning the propagation of goods (such as CSP reduction operators =-=[33]-=-) may provide empirical evidence for it. 21 7 Macroscopic performance of MAC When analysing the performance of MAC at the population level, we are aiming firstly to gauge the general behaviour of the ... |

8 |
search of exceptionally difficult constraint satisfaction problems
- In
- 1994
(Show Context)
Citation Context ...g in the sample at the same constraint tightness, and much more difficult (again by at least an order of magnitude) than 99% of the sample problems in the phase transition. As in our previous studies =-=[36, 39]-=-, we found no ehps in the easy-soluble regions that are insoluble problems: we continue to conjecture that in the case of CSPs such problems must be exceptionally rare 14 , even among ehps. Figure 6 s... |

8 | On the inherent level of local consistency in constraint networks
- Beek
- 1994
(Show Context)
Citation Context ...ose to preprocess problems with AC-3 before search by either MAC or MAC-CBJ 13 . Theoretical work on predicting the existing level of consistency in constraint networks has been presented by van Beek =-=[43]-=-, and empirical studies of AC preprocessing by Borrett & Tsang [4] has shown that its usefulness is restricted to problems that are over-constrained. A transition has been observed between regions whe... |

7 |
The satisfiability constraint gap. Research Paper 702
- Gent, Walsh
- 1994
(Show Context)
Citation Context ... the subproblem is insoluble. We will investigate this belief in Section 8. The results of this study of AC preprocessing may be relevant to the notion of a "constraint gap", proposed by Gen=-=t & Walsh [15]-=- as the conditions arising in sparsely constrained problem classes that give rise to the occurrence of ehps. They show that in SAT problems, ehps tend to occur in regions of the control parameter rang... |

5 | In search of Exceptionally Difficult Constraint Satisfaction Problems - Smith - 1994 |

4 |
Computational Phase Transitions from Real ProbleMS
- Gent, Walsh
- 1995
(Show Context)
Citation Context ...ase transition behaviour has been reported in an increasing number of classes of problem, including satisfiability problems [6, 14, 22, 25], Hamiltonian paths [5], and the travelling salesman problem =-=[16, 17]-=-. Williams and 2 Hogg [45, 46, 47] have developed approximations to the cost of finding the first solution and to the probability that a problem is soluble, both for specific classes of constraint sat... |

2 | Observations on the usefulness of arc consistency preprocessing
- Borrett, Tsang
- 1995
(Show Context)
Citation Context ... MAC-CBJ 13 . Theoretical work on predicting the existing level of consistency in constraint networks has been presented by van Beek [43], and empirical studies of AC preprocessing by Borrett & Tsang =-=[4]-=- has shown that its usefulness is restricted to problems that are over-constrained. A transition has been observed between regions where constraints are very tight and preprocessing eliminates the ent... |