## An Empirical Study of Dynamic Variable Ordering Heuristics for the Constraint Satisfaction Problem (1996)

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Venue: | In Proceedings of CP-96 |

Citations: | 69 - 15 self |

### BibTeX

@INPROCEEDINGS{Gent96anempirical,

author = {Ian P. Gent and Ewan MacIntyre and Patrick Prosser and Barbara M. Smith and Toby Walsh},

title = {An Empirical Study of Dynamic Variable Ordering Heuristics for the Constraint Satisfaction Problem},

booktitle = {In Proceedings of CP-96},

year = {1996},

pages = {179--193},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

. The constraint satisfaction community has developed a number of heuristics for variable ordering during backtracking search. For example, in conjunction with algorithms which check forwards, the Fail-First (FF) and Brelaz (Bz) heuristics are cheap to evaluate and are generally considered to be very effective. Recent work to understand phase transitions in NP-complete problem classes enables us to compare such heuristics over a large range of different kinds of problems. Furthermore, we are now able to start to understand the reasons for the success, and therefore also the failure, of heuristics, and to introduce new heuristics which achieve the successes and avoid the failures. In this paper, we present a comparison of the Bz and FF heuristics in forward checking algorithms applied to randomlygenerated binary CSP's. We also introduce new and very general heuristics and present an extensive study of these. These new heuristics are usually as good as or better than Bz and FF, and we id...

### Citations

820 | Foundations of Constraint Satisfaction
- Tsang
- 1993
(Show Context)
Citation Context ... use variable ordering heuristics that exploit topological parameters, such as width, induced width or bandwidth, and correspond to static instantiation orders (i.e. they do not change during search) =-=[21]-=-. Algorithms that check forwards have additional information at their disposal, such as the current size of the domains of variables. Furthermore, since domain sizes may vary during the search process... |

578 | Where the really hard problems are
- Cheeseman, Kanefsky, et al.
- 1991
(Show Context)
Citation Context ...ny NP-complete problems display a transition in solubility as we increase the constrainedness of problem instances. This phase transition is associated with problems which are typically hard to solve =-=[2]-=-. Under-constrained problems tend to have many solutions and it is usually easy to guess one. Over-constrained problems tend not to have solutions, and it usually easy to rule out all possible solutio... |

459 |
Increasing Tree Search Efficiency for Constraint Satisfaction Problems
- Haralick, Elliott
- 1980
(Show Context)
Citation Context ...jumping (CBJ), and dynamic backtracking (DB) are algorithms that check backwards [11, 5, 6, 10], whereas forward checking (FC) and maintaining arc-consistency (MAC) are algorithms that check forwards =-=[13, 18]-=-. This study investigates only forward checking algorithms, and in particular forward checking combined with conflict-directed backjumping (FC-CBJ) [15]. Algorithm FC instantiates variable v i with a ... |

361 | Dynamic backtracking
- Ginsberg
- 1993
(Show Context)
Citation Context ...pect to that instantiation. Chronological backtracking (BT), backmarking (BM), backjumping (BJ), conflict-directed backjumping (CBJ), and dynamic backtracking (DB) are algorithms that check backwards =-=[11, 5, 6, 10]-=-, whereas forward checking (FC) and maintaining arc-consistency (MAC) are algorithms that check forwards [13, 18]. This study investigates only forward checking algorithms, and in particular forward c... |

350 |
New Methods to Color Vertices of a Graph
- BrÃ©laz
- 1979
(Show Context)
Citation Context ...future variables with v i removed, and is the same as selecting the variable v i which maximizes the denominator of equation (5). 4.2 Heuristic Bz The Brelaz heuristic (Bz) comes from graph colouring =-=[1]-=-; we wish to find a colouring of the vertices of a graph such that adjacent vertices have different colours. Given a partial colouring of a graph, the saturation of a vertex is the number of different... |

350 | Hybrid algorithms for the constraint satisfaction problem
- Prosser
- 1993
(Show Context)
Citation Context ...cy (MAC) are algorithms that check forwards [13, 18]. This study investigates only forward checking algorithms, and in particular forward checking combined with conflict-directed backjumping (FC-CBJ) =-=[15]-=-. Algorithm FC instantiates variable v i with a value x i and removes from the domains of future variables any values that are inconsistent with respect to that instantiation. If the instantiation res... |

204 | Contradicting conventional wisdom in constraint satisfaction
- Sabin, Freuder
- 1994
(Show Context)
Citation Context ...jumping (CBJ), and dynamic backtracking (DB) are algorithms that check backwards [11, 5, 6, 10], whereas forward checking (FC) and maintaining arc-consistency (MAC) are algorithms that check forwards =-=[13, 18]-=-. This study investigates only forward checking algorithms, and in particular forward checking combined with conflict-directed backjumping (FC-CBJ) [15]. Algorithm FC instantiates variable v i with a ... |

175 | Performance Measurement and Analysis of Certain Search Algorithms
- Gaschnig
- 1979
(Show Context)
Citation Context ...pect to that instantiation. Chronological backtracking (BT), backmarking (BM), backjumping (BJ), conflict-directed backjumping (CBJ), and dynamic backtracking (DB) are algorithms that check backwards =-=[11, 5, 6, 10]-=-, whereas forward checking (FC) and maintaining arc-consistency (MAC) are algorithms that check forwards [13, 18]. This study investigates only forward checking algorithms, and in particular forward c... |

116 | The constrainedness of search
- Gent, MacIntyre, et al.
- 1996
(Show Context)
Citation Context ...straint tightness. One simple measure of constrainedness can be derived from the size of problems in the ensemble. Size is determined by both the number of variables and their domain sizes. Following =-=[7, 8]-=-, we measure problem size via the size of the state space being explored. This consists of all possible assignments of values to variables, its size is simply the product of the domain sizes, Q v2V m ... |

113 |
An empirical study of phase transitions in binary constraint satisfaction problems
- Prosser
- 1996
(Show Context)
Citation Context ... Problems from the phase transition are often used to benchmark CSP and satisfiability procedures [22, 9]. Constrainedness can be used both to predict the position of a phase transition in solubility =-=[23, 20, 16, 7, 19]-=- and, as we show later, to motivate the construction of heuristics. In this section, we identify four measures of some aspect of constrainedness. These measures all apply to an ensemble of random prob... |

112 | Locating the phase transition in binary constraint satisfaction problems
- Smith, Dyer
- 1996
(Show Context)
Citation Context ... Problems from the phase transition are often used to benchmark CSP and satisfiability procedures [22, 9]. Constrainedness can be used both to predict the position of a phase transition in solubility =-=[23, 20, 16, 7, 19]-=- and, as we show later, to motivate the construction of heuristics. In this section, we identify four measures of some aspect of constrainedness. These measures all apply to an ensemble of random prob... |

89 |
Backtrack programming
- Golomb, Baumert
- 1965
(Show Context)
Citation Context ...pect to that instantiation. Chronological backtracking (BT), backmarking (BM), backjumping (BJ), conflict-directed backjumping (CBJ), and dynamic backtracking (DB) are algorithms that check backwards =-=[11, 5, 6, 10]-=-, whereas forward checking (FC) and maintaining arc-consistency (MAC) are algorithms that check forwards [13, 18]. This study investigates only forward checking algorithms, and in particular forward c... |

80 |
Search rearrangement backtracking and polynomial average time
- Purdom
- 1983
(Show Context)
Citation Context ...uch as the current size of the domains of variables. Furthermore, since domain sizes may vary during the search process, forward checking algorithms may use dynamic variable ordering (dvo) heuristics =-=[17]-=-, and it is this class of heuristics that is investigated here. 3 Constrainedness Many NP-complete problems display a transition in solubility as we increase the constrainedness of problem instances. ... |

77 | Easy problems are sometimes hard
- Gent, Walsh
- 1994
(Show Context)
Citation Context ...ned". Such problems are usually difficult to solve as they are neither obviously soluble or insoluble. Problems from the phase transition are often used to benchmark CSP and satisfiability proced=-=ures [22, 9]-=-. Constrainedness can be used both to predict the position of a phase transition in solubility [23, 20, 16, 7, 19] and, as we show later, to motivate the construction of heuristics. In this section, w... |

76 |
Phase transition and the mushy region in constraint satisfaction problems
- Smith
- 1994
(Show Context)
Citation Context ...ns for their size. They are therefore likely to be over-constrained and insoluble. A phase transition in solubility occurs inbetween wheres1 [9]. This is equivalent for CSPs to the prediction made in =-=[18]-=- that a phase transition occurs when E(N)s1. 4 Heuristics for Constrainedness Many heuristics in CSPs and other combinatorial problems branch on what can often be seen as an estimate of the most const... |

72 | Exploiting the deep structure of constraint problems
- Williams, Hogg
- 1994
(Show Context)
Citation Context ... Problems from the phase transition are often used to benchmark CSP and satisfiability procedures [22, 9]. Constrainedness can be used both to predict the position of a phase transition in solubility =-=[23, 20, 16, 7, 19]-=- and, as we show later, to motivate the construction of heuristics. In this section, we identify four measures of some aspect of constrainedness. These measures all apply to an ensemble of random prob... |

71 |
Binary Constraint Satisfaction Problems: Some are Harder than Others
- Prosser
- 1994
(Show Context)
Citation Context ...rom the phase transition are now routinely used to benchmark CSP and satisfiability procedures [21, 4, 7]. Constrainedness can be used both to predict the position of a phase transition in solubility =-=[22, 19, 15, 8]-=- and, as we show in the next section, to motivate the construction of heuristics. In this section, we identify four measures which compute some aspect of constrainedness. All of these measures compute... |

53 | A General Backtrack Algorithm That Eliminates Most Redundant Tests
- Gaschnig
- 1977
(Show Context)
Citation Context |

50 |
Consistent-Labeling Problems and Their Algorithms: Expected Complexities and Theory Based Heuristics
- Nudel
- 1983
(Show Context)
Citation Context ... worse than in the previous problem classes. This seems to reaffirm the worth of exploiting information on domain sizes. 6 Discussion Theory-based heuristics for the binary CSP are presented by Nudel =-=[14]-=-, based on the minimization of a complexity estimate, namely the number of compound 100 1000 10000 100000 1e+06 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Kappa Rho FF E(N) Kappa Bz Fig. 4. Performance of FC-CBJ, wi... |

45 |
Domain filtering can degrade intelligent backtracking
- Prosser
- 1993
(Show Context)
Citation Context ...ion. Chronological backtracking (BT), backmarking (BM), backjumping (BJ), conflict-directed backjumping (CBJ), and dynamic backtracking (DB) might all be considered as algorithms that check backwards =-=[11, 5, 6, 13, 10]-=-, whereas forward checking (FC) and maintaining arc-consistency (MAC) might be considered as algorithms that check forwards [12, 17]. This study investigates only forward checking algorithms, and in p... |

42 | Sparse constraint graphs and exceptionally hard problems
- Smith, Grant
- 1995
(Show Context)
Citation Context |

38 |
Experimental evaluation of preprocessing algorithms for constraint satisfaction problems
- Dcchter, Meiri
- 1994
(Show Context)
Citation Context ... is a choice of variable and value to assign to that variable. The order in which decisions are made can have a profound effect on search effort. Dechter and Meiri's study of preprocessing techniques =-=[3]-=- shows that dynamic search rearrangement (DSR), i.e. a variable ordering heuristic that selects as next variable ? This research was supported by HCM personal fellowship to the last author, by a Unive... |

29 | In search of the best constraint satisfaction search
- Frost, Dechter
- 1994
(Show Context)
Citation Context ...ch problems are usually difficult to solve as they are neither obviously soluble or insoluble. Problems from the phase transition are now routinely used to benchmark CSP and satisfiability procedures =-=[21, 4, 7]-=-. Constrainedness can be used both to predict the position of a phase transition in solubility [22, 19, 15, 8] and, as we show in the next section, to motivate the construction of heuristics. In this ... |

27 | Scaling effects in the CSP phase transition
- Gent, MacIntyre, et al.
- 1995
(Show Context)
Citation Context |

24 | B.M.: The phase transition behaviour of maintaining arc consistency
- Grant, Smith
- 1996
(Show Context)
Citation Context ...ant the degree fl of a variable (i.e. the number of constraints incident on a variable) also increases. To avoid this, we vary p 1 with n such that average degree fl remains constant at 5, similar to =-=[12]-=-. To observe a phase transition, 1,000 problems were then generated at each possible value of p 2 from 0:01 to 0:99 in steps of 0:01. In Figure 2, we plot the performance of each heuristic as we incre... |

18 | An attempt to map the performance of a range of algorithm and heuristic combinations
- Tsang, Borrett, et al.
- 1995
(Show Context)
Citation Context ...tics, derived as a result of our studies of phase transition phenomena of combinatorial problems, and compare these against two existing heuristics. Tsang, Borrett, and Kwan's study of CSP algorithms =-=[22]-=- shows that there does not appear to be a universally best algorithm, and that certain algorithms may be preferred under certain circumstances. We carry out a similar investigation with respect to dvo... |

4 |
Interactive scheduling as a constraint satisfaction problem
- Golumbic, Feldman
- 1990
(Show Context)
Citation Context ...s certain simplifications, both approaches can result in the same heuristic such as FF. However, the detailed relationship between the approaches has not yet been fully analysed. Feldman and Golumbic =-=[4]-=- applied Nudel's heuristics to real-world constraint satisfaction problems. Three heuristics are presented, one for a backward checking algorithm (BT), and two for a forward checking algorithm (FC1 an... |