I. P. Gent, E. MacIntyre, P. Prosser, B. M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proc. of CP-96, pp. 179-193.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....size of the D sets to determine the next variable. An example is given in the selectVariable subprocedure in Figure 13. This routine is particularly simple in that it relies purely on the size of the smallest domain, and breaks ties arbitrarily. More sophisticated DVO schemes have been proposed [37,36,74]. backtracking38 with lookahead can be modi ed to employ dynamic variable ordering by calling selectVariable after the initialization step i 1 and after the forward step i i 1. The fc cbj algorithm discussed below illustrates how selectVariable should be invoked. Example 16. Consider ....

I. P. Gent, E. MacIntyre, P. Prosser, B. M. Smith, and T. Walsh. An Empirical Study of Dynamic Variable Ordering Heuristics for the Constraint Satisfaction Problem. In Principles and Practice of Constraint Programming - CP95, pages 179-193, 1996.

.... on N to define orders on the variables. In the remainder of this report we shall assume that #(v) #(w) is true if and only if v is lexicographically smaller than w. The well known minimum domain size heuristic with a lexicographical tie breaker is given by . The Brelaz heuristic [Gent et al. 1996] with a lexicographical tie breaker is given by . An ordering on the maximum original degree with a lexicographical tie breaker is given by o . Note that we only need one of the orders and because a b ( a b. With this equivalence the Brelaz heuristic with a ....

I.P. Gent, MacIntyre E., P. Prosser, B.M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In E.C. Freuder, editor, Principles and Practice of Constraint Programming, pages 179--193. Springer, 1996.

....1 i n. We are now in a position where we need no more notation. For example, the minimum domain size heuristic with a lexicographical tie breaker is given by , the ordering on the maximum original degree with a lexicographical tie breaker is given by o , the Brelaz heuristic (cf. [Gent et al. 1996]) with a lexicographical tie breaker is given by c , and is the lexicographical arc heuristic. As usual, denotes function composition. Related Literature In 1977, Mackworth presented an arc consistency algorithm called AC 3 [Mackworth, 1977] AC 3 has a O(ed ) bound for its ....

I.P. Gent, MacIntyre E., P. Prosser, B.M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In E.C. Freuder, editor, Principles and Practice of Constraint Programming, pages 179--193. Springer, 1996.

....there is another kind of DVOs which it is worth mentioning: those taking into account the tightness of the constraints. They look how much a given constraint restricts the remaining of the problem instead of considering all of them as equivalent. Examples of such heuristics can be found in [11, 7, 8]. The drawback of these heuristics is the need of checking the tightness of a constraint, which is very constraint checks consuming (and thus in cpu time also) None of these heuristics has been shown as better than those dealing only with the constraint graph structure. 2.3 A few words about ....

I.P. Gent, E. MacIntyre, P. Prosser, B. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings CP'96, pages 179193, Cambridge MA, 1996.

....For the rest of this paper, we concentrate on how variable ordering can be applied to a KrustTool. A CSP variable is generally constrained in two ways, rstly by the constraints it is involved in and secondly by its domain size. Most common variable ordering heuristics exploit these 2 properties [5, 8]. Heuristics for static ordering exploit relationships among variables identi ed from the topology of the constraint graph [17] Dynamic variable ordering addresses the fact that invariably the best variable order is di erent in di erent branches of the search tree, by taking advantage of the ....

Ian Gent, Ewan MacIntyre, Patrick Prosser, Barbara Smith, and Toby Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In in Principles and Practice of Constraint Programming, pages 179-193. Springer-Verlag, 1996.

....of which the first has duration 2 time units and starting time [0. 2 4] the second duration 1 and starting time [8. 9] and the third duration 3 and starting time [12] then the array LectureTimeUnits will have 6 elements (the sum of the lectures durations) which will be [0. 2 4] 1. 3 5] 8. 9] 12][13][14] The array TimeTable has D H elements which correspond to the D H time units of a timetable. Each one of them takes values within the interval [ 1. d] where d is the number of elements in array LectureTimeUnits. When all lectures have been added, then the Solver function inverse is called. ....

....how that affects their ability to find one. 6.2 Heuristics The celebrated First Fail [17] and Brelaz [4] variable ordering heuristics are employed in our implementation. Also, some other general purpose heuristics, the kappa family of heuristics for constraint satisfaction problems, proposed in [13], are also included. The First Fail selection criterion is supposed to follow the rule in order to succeed, try first where you are most likely to fail . Although it is argued that selecting the variable with the least values in its domain leads to harder subproblems, this heuristic is efficient ....

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I. P. Gent, E. MacIntyre, P. Prosser, B. M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of the 2nd International Conference on the Principles and Practice of Constraint Programming CP '96, pages 179--193, 1996.

....program, but it is guaranteed to make the program run, for any instance, almost exactly) as fast as the fastest heuristic for that instance. From the results of the empirical study, we can also conclude the following, regarding subset decision problems: ffl As instances get less constrained [5], the default VVO heuristic almost always performs best. ffl As instances get more constrained, the performance of the default VVO heuristic degenerates (see Figure 3) ffl As instances get more constrained, the static and dynamic VVO heuristics behave much more gracefully, rather than seeing ....

I.P. Gent, E. MacIntyre, P. Prosser, B.M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proc. of CP'96, pp. 179-- 193. The MIT Press, 1996.

....et al. 1991) Crawford and Auton (1993) Mitchell et al. 1992) Kirkpatrick and Selman (1994) and Smith and Dyer (1996) Hogg et al. 1996) contains a collection of recent papers in the area. Some important related work on hard problem instances involving some underlying structure is that of Gent and Walsh (1995) and Zhang and Korf (1996). Both teams use the structure of combinatorial optimization problems, such as the Traveling Salesman Problem and real world Timetabling problems. By varying the constraint density of their problem instances they obtain varying degrees of difficulty. One important difference is that in our ....

Gent, I.P. , MacIntyre, E., Prosser, Smith, B., and Walsh, T. (1996) An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. Proc. CP-96, Boston, MA, 179--193.

....involve an interchange between variables and values. Car sequencing is a relatively straightforward problem, and we expect that we will be able to perform both a theoretical and empirical study of this problem. In particular, we aim to derive theory based heuristics for this problem (as in [2, 3]) The latter part of the work investigates the job shop scheduling problem (JSSP) and the vehicle routing problem (VRP) Reformulation will involve mapping one problem to the other, i.e. mapping from one domain to another. Therefore, we investigate two classes of reformulation: the first within a ....

....path. Objectives and Milestones Since the car sequencing problem is relatively straight forwards (when compared to the JSSP and VRP in Work Package 2) our first objective will be a formal definition of constrainedness [2] for this problem and the derivation of heuristics based on this [3]. Our second objective is to provide guidance on the relationship between representations and heuristics in assembly line problems. Our first milestone is an experimental laboratory for car sequencing. We will have constraint based models for the various representations and problem generators. The ....

I.P. Gent, E. MacIntyre, P. Prosser, B.M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of CP-96, pages 179--193. Springer, 1996.

....heuristics are used to reduce the size of the search tree. Value ordering heuristics are used to guide the search to a solution. In the case of branch and bound this will tighten the bound limit early, and make the remainder of search faster. It is generally believed, and recently been shown [9] that an effective variable ordering strategy is to instantiate more constrained variables first. Of course, constrainedness is a problem dependent attribute, but some mathematical descriptions of it can be found for simple random problems [10, 11] In our context, a variable (visit) could be ....

....There have been advances in recent years in the efficiency of propagation algorithms. This has been both in the general case [2, 3, 26] and in the case of specific common types of constraint, for example [22, 23] There has also been progress made on variable ordering heuristics both for search [9], and for constraint propagation algorithms [8] Although our method works extremely well (see results in section 3) it can still be combined with minima escaping procedures if desired. Techniques such as simulated annealing and tabu search could be applied, but the fit is not good as it s not ....

I. Gent, E. MacIntyre, P. Prosser, B. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of CP-96. Springer, 1996.

....of branch and bound can be increased by the use of variable and value ordering heuristics. Variable ordering heuristics are used to reduce the size of the search tree. Value ordering heuristics are used to guide the search to a solution. It is generally believed, and experimental evidence suggests [9, 10] that an effective variable ordering strategy is to instantiate more constrained variables first. In our context, a variable (visit) could be considered constrained if it is far removed from the rest of the routing plan (and so will constrain the plan when inserted, due to high consumption of ....

I. Gent, E. MacIntyre, P. Prosser, B. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of CP-96. Springer, 1996.

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I.P. Gent, E. MacIntyre, P. Prosser, B.M. Smith, and T. Walsh, `An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem', Principles and Practice of Constraint ProgrammingCP '96, 179--193, (1996).

.... 8i, h32; 8i no [ECF,CDE] 13] B h100; 6i no [IPG,EM,PP,TW] 27] B h20; 10i no [KK,RD] 36] B h100; 8i, h125; 6i, h150; 4i no CP 96 [CB,JCR] 4] B h35; 6i p2 = 4=36 h125; 3i, h350; 3i p2 = 1=9 h35; 9i, h50; 6i, h50; 20i, no [DAC,JF,IPG,EM,NT,TW] 9] B h20; 10i no [IPG,EM,PP,BMS,TW] [24] B h20 50; 10i no [JL,PM] 40] B h15; 5i p2 = 1=25 4=25 h10; 10i p2 = 1=100 9=100 [RJW] 57] A h30; 5i, h100; 5i no ECAI 96 [JEB,EPKT,NRW] 5] B h50; 10i no [BC,GV,DM,PB] 6] B h50; 10i, h20; 5i no [SAG,BMS] 34] B h30 70; 10i no [ACMK,EPKT,JEB] 38] B h30; 5i p2 = 0:12 h40; 5i p2 = 0:08 ....

....in the size of the problem, so we were able to experiment with problems containing thousands of variables with large samples. Since awed variables are more likely in dense constraint graphs, we generated problems with complete constraint graphs (i.e. with p 1 = 1) As in other studies (e.g. [33, 24]) we also generated a separate ensemble of problems in which the constraint graph has constant average degree, That is, p 1 = n 1) The constraint tightness for which the expected number of solutions is 1 is then constant as n increases; this constraint tightness is often a good predictor ....

I.P. Gent, E. MacIntyre, P. Prosser, B.M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of CP-96, pages 179-193. Springer, 1996.

....of the constraint graph. Another heuristic which addresses the same difficulty is dom deg, introduced by Bessi ere and R egin [1] which chooses the variable minimizing the ratio of current domain size to (original) degree. A number of dynamic variable ordering heuristics were introduced in [5], based on the principle of minimizing the constrainedness of the subproblem consisting of the future variables and their remaining values. These heuristics were compared with the Br elaz heuristic: the Br elaz heuristic was found to be the best of those studied on uniform binary problems with low ....

....with the Br elaz heuristic: the Br elaz heuristic was found to be the best of those studied on uniform binary problems with low constraint density, that is problems with uniform domain sizes and constraint tightnesses, of the kind described below in section 3. Two of the heuristics 2 proposed in [5], however, have lower search costs than the Br elaz heuristic when the constraint graph is complete. 3 The Random CSP Model Nudel s theory (outlined below) allows us to calculate the expected number of nodes visited in solving certain kinds of CSP. In order to apply the theory, we need a ....

I. Gent, E. MacIntyre, P. Prosser, B. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In E. C. Freuder, editor, Principles and Practice of Constraint Programming - CP96, LNCS 1118, pages 179--193. Springer-Verlag, 1996.

.... 8i, h32; 8i no [ECF,CDE] 13] B h100; 6i no [IPG,EM,PP,TW] 27] B h20; 10i no [KK,RD] 36] B h100; 8i, h125; 6i, h150; 4i no CP 96 [CB,JCR] 4] B h35; 6i p2 = 4=36 h125; 3i, h350; 3i p2 = 1=9 h35; 9i, h50; 6i, h50; 20i, no [DAC,JF,IPG,EM,NT,TW] 9] B h20; 10i no [IPG,EM,PP,BMS,TW] [24] B h20 Gamma 50; 10i no [JL,PM] 40] B h15; 5i p2 = 1=25 Gamma 4=25 h10; 10i p2 = 1=100 Gamma 9=100 [RJW] 56] A h30; 5i, h100; 5i no ECAI 96 [JEB,EPKT,NRW] 5] B h50; 10i no [BC,GV,DM,PB] 6] B h50; 10i, h20; 5i no [SAG,BMS] 34] B h30 Gamma 70; 10i no [ACMK,EPKT,JEB] 38] B h30; 5i p2 ....

....linear in the size of problems, so we were able to experiment with problems containing thousands of variables with large samples. Since flawed variables are more likely in dense constraint graphs, we generated problems with complete constraint graphs (i.e. with p 1 = 1) As in other studies (e.g. [33, 24]) we also generated a separate ensemble of problems in which the average degree of the vertices in the constraint graph is kept constant. That is, we vary p 1 as 1= n Gamma1) As we argue in Section 9, the constraint tightness at the phase transition then remains roughly constant. Keeping the ....

I.P. Gent, E. MacIntyre, P. Prosser, B.M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of CP-96, pages 179--193. Springer, 1996.

....minimize kappa heuristic Figure 9. Fail First (FF) and minimize heuristics on h20; 10; p 1 ; p 2 i problems using FC CBJ. Mean consistency checks on y axis, and the constrainedness of problems, on 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 t = 46:3 at p 1 = 1:0, all in favour of minimize . To check the validity of these values we ....

....the expected number of solutions, hSoli [50] Given a choice of two subproblems with equal hSoli, the heuristic of minimizing will branch into the smaller problem in the expectation 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 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 is best to branch ....

I.P. Gent, E. MacIntyre, P. Prosser, B.M.Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of CP-96, pages 179--193. Springer Verlag, 1996.

....that a variable is involved in, as well as their number, clearly has an influence on the difficulty of assigning a value to it, this should also be taken into account. The theory on which this heuristic was based was developed from an investigation of variable ordering heuristics, described in [5]. A number of heuristics which take into account the tightness of the constraints that a variable is involved in were proposed; in the particular case when all variables have the same domain size (which is true in this case, if we are looking for an initial ordering of the cars) several of these ....

I. Gent, E. MacIntyre, P. Prosser, B. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of CP'96, pages 179--193, Aug. 1996.

....made. However, you should collect as many meaningful aspects of data as you can think of. For a long time, we did not record the number of branching points for backtracking procedures, a statistic subtly different from the number of branches. In investigating the satisfiability constraint gap [Gent and Walsh 1996], the most meaningful statistic turned out to be the ratio of constraint propagations to branching points. We therefore had to rerun many experiments. To have produced this data in the first place would have involved almost no extra expense. It pays to collect everything you can think of, whether ....

....would vary outside our control. Controlling sources of variation is not the same as eliminating them. We later wished to experiment on non uniform problems in which p 2 varies within a problem. To do this we designed a random problem generator to allow for this explicitly and under our control [Gent et al. 1996]. 5 Analysis of Data Having run your computational experiments, you are now in a position to analyse the data. Somewhat surprisingly, it is often quite hard to determine the actual outcome of the experiments. One reason is the mega bytes of data generated. Here are some of the lessons we have ....

I. Gent, E. MacIntyre, P. Prosser, B. Smith and T. Walsh. 1996. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of CP-96, 179--193. Springer.

.... B h16; 8i, h32; 8i no [ECF,CDE] 12] B h100; 6i no [IPG,EM,PP,TW] 25] B h20; 10i no [KK,RD] 34] B h100; 8i, h125; 6i, h150;4i no CP 96 [CB,JCR] 4] B h35; 6i p2 = 4=36 h125; 3i, h350; 3i p2 = 1=9 h35; 9i, h50; 6i, h50; 20i, no [DAC,JF,IPG,EM,NT,TW] 9] B h20; 10i no [IPG,EM,PP,BMS,TW] [23] B h20 Gamma 50;10i no [JL,PM] 38] B h15; 5i p2 = 1=25 Gamma 4=25 h10; 10i p2 = 1=100 Gamma 9=100 [RJW] 52] A h30; 5i, h100;5i p2 = 0:1 ECAI 96 [JEB,EPKT,NRW] 5] B h50; 10i no [BC,GV,DM,PB] 6] B h50; 10i, h20; 5i no [SAG,BMS] 32] B h30 Gamma 70;10i no [ACMK,EPKT,JEB] 36] B h30; 5i p2 = ....

....linear in the size of problems, so we were able to experiment with problems containing thousands of variables with large samples. Since flawed variables are more likely in dense constraint graphs, we generated problems with complete constraint graphs (i.e. with p 1 = 1) As in other studies (e.g. [31, 23]) we also generated a separate ensemble of problems in which the average degree of the vertices in the constraint graph is kept constant. That is, we vary p 1 as 1= n Gamma 1) As we argue in Section 9, the constraint tightness at the phase transition then remains roughly constant. Keeping the ....

I.P. Gent, E. MacIntyre, P. Prosser, B.M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of CP-96, pages 179--193. Springer, 1996.

....although this might give good results for hn; m; p 1 ; p 2 i problems, it requires subverting the fail first principle and would give no guidance in cases where smallest remaining domain is not a good heuristic. An alternative principle for variable ordering heuristics is proposed by Gent et al. [5], namely that the next variable should be chosen so as to minimize the constrainedness of the future subproblem. The hope is that this will guide the search towards under constrained subproblems, since underconstrained problems tend to have many solutions and be easy to solve. Four new heuristics ....

I. Gent, E. MacIntyre, P. Prosser, B. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In E. C. Freuder, editor, Principles and Practice of Constraint Programming - CP96, volume 1118 of Lecture Notes in Computer Science, pages 179--193. Springer-Verlag, Aug. 1996.

.... for the constrainedness of computational problems that predicts the location and shape of such phase transitions [10] Finally, in collaboration with Barbara Smith and Ewan MacIntyre, they have proposed several new heuristics for constraint satisfaction problems based upon this definition [8]. Track record: From 1987 to 1989, Patrick Prosser led the Alvey funded project Application of Artificial Intelligence to the management of heavy manufacturing industry, with Alcan Plate as a collaborator and end user. This resulted in the development of a novel distributed asynchronous ....

.... with Ewan MacIntyre, they have proposed an unifying definition for the constrainedness of computational problems [10] Finally, in collaboration with Barbara Smith and Ewan MacIntyre, they have proposed several new heuristics for constraint satisfaction problems based upon this definition [8]. Originality: Research in phase transition phenomena has progressed in different areas independently. This proposal attempts to unify such research across a wide variety of problem domains. The proposal is also very timely. Similar results have been seen in different problem domains. We are ....

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I.P. Gent, E. MacIntyre, P. Prosser, B. Smith and T. Walsh. An Empirical Study of Dynamic Variable Ordering Heuristics for the Constraint Satisfaction Problem. To appear in Proceedings of Second International Conference on Principles and Practice of Constraint Programming, Cambridge, Massachusetts, 1996.

....constraint satisfaction problems (Gent et al. 1995) The phase transition again occurs around 1. In number partitioning, we have n numbers from the range (0; l] and wish to find an exact partition into m bags with the same sum. We have N = n log 2 m as there are m n possible partitions. (Gent Walsh 1996) present an annealed theory in which the expected number of exact partitions is hSoli m n ( 1 2 ) m Gamma1) log 2 (l) Although (2) no longer applies, substituting N and hSoli into (1) gives = m Gamma 1) log m (l) n (3) In two way partitioning, i.e. m = 2, a phase transition in ....

.... in which the expected number of exact partitions is hSoli m n ( 1 2 ) m Gamma1) log 2 (l) Although (2) no longer applies, substituting N and hSoli into (1) gives = m Gamma 1) log m (l) n (3) In two way partitioning, i.e. m = 2, a phase transition in solubility occurs at = 0:96 (Gent Walsh 1996). We thus see that our definition of generalises a number of parameters introduced in a variety of problem classes. This suggests that constrainedness is a fundamental property of problem ensembles. In addition to unifying existing parameters, we can now compare problems between classes. For ....

[Article contains additional citation context not shown here]

Gent, I.; MacIntyre, E.; Prosser, P.; Smith, B.; and Walsh, T. 1996. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. Research Report 96.05, School of Computer Studies, University of Leeds.

....first moment bound, we generated random problems from the popular model B and calculated the fraction with a flawed variable. Since flawed variables are more likely in dense constraint graphs, we generated problems with complete constraint graphs (i.e. with p 1 = 1) As in other studies (e.g. [12, 6]) we also generated a separate set of problems in which the average degree of the vertices in the constraint graph is kept constant. That is, we vary p 1 as 1= n Gamma 1) As we argue in Section 9, the constraint tightness at the phase transition then remains roughly constant. Keeping the ....

I.P. Gent, E. MacIntyre, P. Prosser, B.M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In 2nd International Conference on Principles and Practices of Constraint Programming (CP-96), pages 179--193, 1996.

....(again in thousands) No claims are drawn from the above results, for example that one heuristic is better than another, because the sample size is too small and the problem data too specific 2 . 2 For example, we will get a different ranking of the heuristics if we vary problem features[3]. The contours in Figure 2 show, for the RAND heuristic, the average number of consistency checks performed at varying depths in the search tree, nodes visited, and variables selected. Note that the y axis is a logscale. The curves look quite natural, with the peak in search effort taking place ....

I.P. Gent, E. MacIntyre, P. Prosser, B.M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for constraint satisfaction problems. In Proc. CP96, pages 179--193, 1996.

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I. P. Gent, E. MacIntyre, P. Prosser, B. M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proc. of CP-96, pp. 179-193.

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I.P. Gent, MacIntyre E., P. Prosser, B.M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In E.C. Freuder, editor, Principles and Practice of Constraint Programming, pages 179--193. Springer, 1996.

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I. Gent, E. MacIntyre, P. Prosser, B. Smith, T. Walsh, An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem, in: Proc. 2nd International Conference on Principles and Practice of Constraint Programming, Cambridge, MA, 1996, pp. 179--193.

No context found.

Gent, I., E. MacIntyre, P. Prosser, B. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem, in CP96. 1996.

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I. Gent, E. MacIntyre, P. Prosser, B. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of the Second International Conference on Principles and Practice of Constraint Programming (CP96), pages 179--193, August 1996.

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Ian P. Gent, Ewan MacIntyre, Patrick Prosser, Barabara M. Smith, and Toby Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Eugene C. Freuder, editor, Second International Conference on Principles and Practice of Constraint Programming, pages 179--193. Springer, 1996.

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I.P. Gent, E. MacIntyre, P. Prosser, B.M. Smith, and T. Walsh. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Freuder [Fre96], pages 179--193.

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