## The Constrainedness of Search (1999)

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Venue: | In Proceedings of AAAI-96 |

Citations: | 116 - 26 self |

### BibTeX

@INPROCEEDINGS{Gent99theconstrainedness,

author = {Ian P. Gent and Patrick Prosser and Toby Walsh},

title = {The Constrainedness of Search},

booktitle = {In Proceedings of AAAI-96},

year = {1999},

pages = {246--252}

}

### Years of Citing Articles

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### Abstract

We propose a definition of `constrainedness' that unifies two of the most common but informal uses of the term. These are that branching heuristics in search algorithms often try to make the most "constrained" choice, and that hard search problems tend to be "critically constrained". Our definition of constrainedness generalizes a number of parameters used to study phase transition behaviour in a wide variety of problem domains. As well as predicting the location of phase transitions in solubility, constrainedness provides insight into why problems at phase transitions tend to be hard to solve. Such problems are on a constrainedness "knife-edge", and we must search deep into the problem before they look more or less soluble. Heuristics that try to get off this knife-edge as quickly as possible by, for example, minimizing the constrainedness are often very effective. We show that heuristics from a wide variety of problem domains can be seen as minimizing the constrainedness (or proxies ...

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Citation Context ...i . This is done for all future variables and the variable with minimum cost is selected as the current variable. We compared the minimize- heuristic with an encoding of the fail first (FF) principle =-=[26]-=- i.e. selecting the variable with smallest domain. Figure 9 shows the results of experiments performed on h20; 10; p 1 ; p 2 i problems (i.e. 20 variables, uniform domain size of 10). Constraint densi... |

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Citation Context ...om the class hn; m; p 1 ; p 2 i described earlier. We encoded minimizingsas a dynamic variable ordering heuristic within the algorithm FC-CBJ (i.e. forward checking with conflict-directed backjumping)=-=[46]-=-. After instantiating a variable, domain filtering is performed. This may result in a reduction in the size of the domains of future (i.e. uninstantiated) variables and consequently alter the tightnes... |

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Citation Context ... intensive study in recent years in a large number of problem domains including, for example, propositional satisfiability, graph colouring, constraint satisfaction problems, and hamiltonian circuits =-=[7, 43, 51, 57]-=-. Here, we introduce some general methods which help to answer these questions in a wide range of problems. These methods are based on a definition of the constrainedness of an ensemble of combinatori... |

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Citation Context ...th the FF heuristic took 61,200 consistency checks to solve this problem, and just 39,177 checks with the minimize- heuristic. We therefore ran our experiments on problems with fewer variables. As in =-=[17]-=- and [20], we used h10; 10; 1; p 2 i problems varying p 2 from 0.2 to 0.5 with a sample size of 50. In the middle of the phase transition, the maximize- heuristic is an order of magnitude worse than t... |

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Citation Context ...ransition has been shown to occur experimentally around l=n = 4:3 [43, 10], which corresponds tos0:82. Theoretical bounds put the phase transition for random 3-Sat in the interval 3:003 ! l=n ! 4:598 =-=[15, 34]-=-. This corresponds to the interval 0:58 !s! 0:89. For random 4-Sat, the phase transition has been shown to occur experimentally around l=n = 9:8 [21], which corresponds tos0:91. For large k, the phase... |

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Citation Context ...pectation that this is less constrained. Experiments so far have failed to show which heuristic, if either, is better [18]. Hooker and Vinay investigate the Jeroslow-Wang heuristic for satisfiability =-=[29]. They propose the &-=-quot;satisfaction hypothesis", that it is best to branch into subproblems that are more likely to be satisfiable, but reject this in favour of the "simplification hypothesis", that it i... |

113 |
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Citation Context ...rating test problems is to have n variables each with the same domain size of m. Given a constraint density of p 1 , exactly p 1 n(n \Gamma 1)=2 constraints 8 are chosen, each with a tightness of p 2 =-=[47, 51]-=-. Such problems are described by the tuple, hn; m; p 1 ; p 2 i. Using these values, (3) givess= n \Gamma 1 2 p 1 log m ( 1 1 \Gamma p 2 ) This has been used as a parameter for binary constraint satisf... |

112 | Locating the phase transition in binary constraint satisfaction problems
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Citation Context ... intensive study in recent years in a large number of problem domains including, for example, propositional satisfiability, graph colouring, constraint satisfaction problems, and hamiltonian circuits =-=[7, 43, 51, 57]-=-. Here, we introduce some general methods which help to answer these questions in a wide range of problems. These methods are based on a definition of the constrainedness of an ensemble of combinatori... |

103 |
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Citation Context ... in many different problem classes as a phase transition in solubility often occurs around a critical value of l=n [43, 21]. For random 2-Sat, the phase transition has been proven to occur at l=n = 1 =-=[8, 25]-=-, which corresponds tos0:42. For random 3-Sat, the phase transition has been shown to occur experimentally around l=n = 4:3 [43, 10], which corresponds tos0:82. Theoretical bounds put the phase transi... |

97 | Tail bounds for occupancy and the satisfiability threshold conjecture. Random Structures and Algorithms
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Citation Context ...es can be achieved either by taking account of the variance in the number of solutions at the phase boundary [57, 51], or by finding an equivalent problem with fewer solutions at its phase transition =-=[32, 13]-=-. There is a subtle difference between the prediction of a phase transition ats1 and at hSolis1. While hSoli at the phase transition can grow exponentially with N , the value ofstends to vary more slo... |

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Citation Context ...gree has been used as an order parameter for describing the phase transition in colouring problems [7]. A phase transition has been observed in random 3-colouring problems at an average degree of 4.6 =-=[28]-=-, corresponding tos= 0:84. In random 4-colouring problems, the phase transition occurs around an average degree of 8.7 [53], corresponding tos= 0:90. In random 5-colouring problems, the phase transiti... |

82 | Approximating the unsatisfiability threshold of random formulas - Kirousis, Kranakis, et al. - 1996 |

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Citation Context ...search effort is measured as the number of trial instantiations of variables, minimize- again shows superior mean and median 1 While technically these instances were subject to the flaw identified by =-=[2]-=-, instances generated with these parameters are never flawed in practice [38]. 21 100 1000 10000 100000 1e+06 0 0.5 1 1.5 2 2.5 FF heuristic minimize kappa heuristic Figure 9. Fail First (FF) and mini... |

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Citation Context ... 1 ; p 2 i problems using FC-CBJ. Mean consistency checks on y-axis, and the constrainedness of problems,son x-axis. Contours for p 1 = 1:0 (top), p 1 = 0:5 (middle), p 1 = 0:2 (bottom). performance. =-=[18]-=- reports more extensive experiments on the minimize- heuristic with similar results. At the peak in search costs, paired-sample t-tests gave values of t = 12:3 at p 1 = 0:2, t = 24:4 at p 1 = 0:5, and... |

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Citation Context ...ated but not identical to both these hypotheses: in general it will seek out simple problems that are likely to be soluble. Nudel has proposed some theoretically motivated heuristics for binary CSP's =-=[45]-=-. Two classes of heuristic are presented, global and local. Global heuristics fix the instantiation order at the start of search, whereas local heuristics take account of information made available du... |

49 | Trying harder to fail first
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Citation Context .... It can therefore be difficult to decide how rigorously to apply a given theory about heuristic construction. Smith and Grant investigated this problem for the `fail first' principle in binary CSP's =-=[52]-=-, a principle often used to justify the minimum domain size (FF) heuristic. They found that a heuristic which did more work to maximize the probability of an early failure did significantly more searc... |

46 |
Branch-and-cut solution of inference problems in propositional logic
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Citation Context ...on of other literals. As empty and unit clauses typically makes problems much easier, Hooker and Fedjki have proposed that such clauses should be discarded and longer clauses generated in their place =-=[30]-=-. Gent and Walsh show that if the expected clause length is kept roughly constant by varying p as 1=n then the solubility phase transition occurs around a constant value of l=n [21]. They approximate ... |

45 | The constrainedness of arc consistency
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Citation Context ...complexity classes. For example, we have modelled phase transition behaviour and suggested new heuristics for polynomial problems like establishing arc consistency in constraint satisfaction problems =-=[19]-=-. Whatever the complexity class, our definition of constrainedness may be able to identify phase transition behaviour and suggest heuristics that help us solve search problems. Musick and Russell mode... |

41 |
Critical behavior in the computational cost of satisfiability testing
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Citation Context ... size scaling we are able to model scaling behaviour of the phase transition. 7 Search cost Modelling using finite size scaling can be applied to measures of algorithmic behaviour such as search cost =-=[49, 20]-=-. For the Atsp, we use the 90th percentile 14 of the number of leaf nodes searched, as lower percentiles such as median cost were always trivial since almost no backtracking occurred. As in many other... |

38 |
Experimental evaluation of preprocessing algorithms for constraint satisfaction problems
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Citation Context ...en future and past variables. We minimize the numerator of (3) by choosing a variable that has most constraints with past variables. This corresponds to the maximum cardinality heuristic described in =-=[12]-=-. We may take advantage of both numerator and denominator of (3). One way to do this is to choose the variable with smallest domain size (maximizing the denominator) and break ties by choosing the tie... |

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Citation Context ...in some sense be more constrained. Computational results appear to support this claim. For example, with problems at the phase transition Ckk, one of the best algorithms known for number partitioning =-=[35]-=-, runs on average as approximately 2 0:85N [24] whilst Crawford's Tableau algorithm, one of the best algorithms known for satisfiability, runs on average as approximately 2 0:05N [11]. The definition ... |

30 | Phase transitions and annealed theories: Number partitioning as a case study
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Citation Context ...e n numbers drawn uniformly and at random from the range (0; l] and wish to find a partition into two bags with the same sum. To study this problem, Gent and Walsh have developed an "annealed&quo=-=t; theory [22, 24]-=- in which they average probabilities independently over the different binary digit positions. They call this an annealed theory by analogy with the annealed theory of materials which averages independ... |

30 |
Critical Behaviour in the satisfiability of random Boolean expressions, Science 264
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Citation Context ...n can grow exponentially with N , the value ofstends to vary more slowly. For example, with random 3Sat problems, the expected number of solutions at the phase boundary grows as approximately 2 0:18N =-=[33]-=-. By comparison,sat the phase boundary tends to vary much more slowly. As we show later, the variation insat the phase boundary can be modelled by the technique of finite size scaling using a low-orde... |

30 | Random constraint satisfaction: theory meets practice
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Citation Context ..., minimize- again shows superior mean and median 1 While technically these instances were subject to the flaw identified by [2], instances generated with these parameters are never flawed in practice =-=[38]-=-. 21 100 1000 10000 100000 1e+06 0 0.5 1 1.5 2 2.5 FF heuristic minimize kappa heuristic Figure 9. Fail First (FF) and minimize- heuristics on h20; 10; p 1 ; p 2 i problems using FC-CBJ. Mean consiste... |

29 |
Finite-size scaling
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Citation Context ....4 1.6 1.8 2 n=48 n=42 n=36 n=30 n=24 n=18 n=12 n= 6 Figure 2. Probability of tour of required length existing in Atsp, plotted againstsfor 6 to 48 cities. described using finite size scaling methods =-=[3]-=-. Around a critical temperature T c , problems of all sizes tend to be indistinguishable except for a change of scale given by a power law in a characteristic length. Here we propose that the constrai... |

29 | Summarizing CSP hardness with continuous probability distributions
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Citation Context ...ysis of behaviour as problem size scales, with analyses which study the distribution of search costs at a single problem size, such as those carried out by Frost, Rish and Villa, and Hoos and Stutzle =-=[16, 48, 31]-=-. 8 Constrainedness within search A general rule of thumb in solving search problems is to tackle the hardest part first. Many heuristics therefore try to branch on the most constrained variable. To t... |

27 | Scaling effects in the CSP phase transition
- Gent, MacIntyre, et al.
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(Show Context)
Citation Context ...are described by the tuple, hn; m; p 1 ; p 2 i. Using these values, (3) givess= n \Gamma 1 2 p 1 log m ( 1 1 \Gamma p 2 ) This has been used as a parameter for binary constraint satisfaction problems =-=[20]-=-. For h20; 10; p 1 ; p 2 i problems, the phase transition occurs between 0:75s1. For hn; 3; p 1 ; 2=9i problems (which resemble 3-colouring problems in having three values), the phase transition occur... |

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(Show Context)
Citation Context ...ysis of behaviour as problem size scales, with analyses which study the distribution of search costs at a single problem size, such as those carried out by Frost, Rish and Villa, and Hoos and Stutzle =-=[16, 48, 31]-=-. 8 Constrainedness within search A general rule of thumb in solving search problems is to tackle the hardest part first. Many heuristics therefore try to branch on the most constrained variable. To t... |

24 | Analysis of heuristics for number partitioning
- Gent, Walsh
- 1998
(Show Context)
Citation Context ...e n numbers drawn uniformly and at random from the range (0; l] and wish to find a partition into two bags with the same sum. To study this problem, Gent and Walsh have developed an "annealed&quo=-=t; theory [22, 24]-=- in which they average probabilities independently over the different binary digit positions. They call this an annealed theory by analogy with the annealed theory of materials which averages independ... |

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- 1998
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Citation Context ... is asymptotically incorrect. It is interesting to speculate that the discrepancy between the two parameters may be related to the fact that hard problems do not seem to occur at the phase transition =-=[55]. 4.6 Comp-=-aring domains We thus see that our definition ofsgeneralizes a number of parameters introduced in a variety of problem classes. This suggests that "constrainedness" is a fundamental property... |

22 |
Some new branching and bounding criteria for the asymmetric traveling salesman problem
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(Show Context)
Citation Context ...lated numerically by computer. We expect a phase transition in the decision problem whens1. We tested this experimentally using a branch and bound algorithm with the Hungarian heuristic for branching =-=[6]-=-. For n=6 to 48, we randomly generated 1000 problems with inter-city distances independently and normally distributed with =10 6 and oe=10 5 . Figure 2 shows the probability that there was a tour less... |

21 |
Phase transition in number partitioning problem
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(Show Context)
Citation Context ...d arounds= 0:96 [22, 24]. Using some complex analysis based on statistical thermodynamics, Mertens predicts the location of this phase transition around a fixed value of a parameter given bys+O(1=nl) =-=[40]-=-. In m-way number partitioning, we have n numbers drawn uniformly and at random from the range (0; l] and wish to find a partition into m bags with the same sum. As there are m n possible partitions o... |

21 | The constrainedness knife-edge
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(Show Context)
Citation Context ...o remain critically constrained. Here we report the results for propositional satisfiability. However, results were similar in the other domains which included graph colouring and number partitioning =-=[56]-=-. We use the Davis-Putnam procedure with unit propagation but no pure literal deletion. We branch with Mom's heuristic, picking the literal that occurs most often in the minimal size clauses, breaking... |

18 | An attempt to map the performance of a range of algorithm and heuristic combinations
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- 1995
(Show Context)
Citation Context ...a sample of 1000 in each case, which never gave a value above t = 3:5. This provides strong statistical evidence that the minimize- heuristic is better than the FF heuristic in these problem classes. =-=[54]-=- give results on the same problem classes seen in Figure 9, on a range of algorithm/heuristic combinations. For high values of p 1 they report that FCCBJ with the FF heuristic was the best combination... |

17 | Integrating heuristics for constraint satisfaction problems: A case study
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- 1993
(Show Context)
Citation Context ...irical approach. For example, the Multi-tac system constructs a set of candidate branching rules and benchmarks them to determine their effectiveness as heuristics on a representative set of problems =-=[41]-=-. Our approach to this problem is more analytical and is motivated by studies of transitions in solubility. By comparing the parameters introduced in a variety of domains like graph colouring and sati... |

17 | Is there any need for domain-dependent control information: a reply
- Minton
- 1996
(Show Context)
Citation Context ...t is instructive to consider what sort of theory would be required to be able to prove that, in any given circumstance, a variable is most likely to be most constraining .." [author's emphasis] p=-=.861 [42]-=- Minton argues that this is too hard a problem to solve analytically, and proposes instead an empirical approach. For example, the Multi-tac system constructs a set of candidate branching rules and be... |

16 | Phase transitions from real computational problems - Gent, Walsh - 1995 |