constraints to the Maximum-Density Still Life and Peaceable Armies of Queens problems. In the latter we are able to improve on known results. In a broader con-.
Using Pseudosymmetry to Reduce Search Effort Steven Prestwich and J. Christopher Beck Cork Constraint Computation Centre Department of Computer Science, University College Cork Cork, Ireland s.prestwich,c.beck @4c.ucc.ie
Abstract. Symmetry breaking exploits a relationship between solutions: that they occur in equivalence classes. Constraints may be added to exclude all symmetrically equivalent solutions but one, greatly reducing the search space. Using two problems we show that a weaker relationship can also be exploited: that the existence of one solution implies the existence of another but not necessarily vice versa. It is then sufficient to search only for the latter solution. We demonstrate the value of using such relationships in search by adding pseudosymmetry breaking constraints to the Maximum-Density Still Life and Peaceable Armies of Queens problems. In the latter we are able to improve on known results. In a broader context, we show that pseudosymmetry is a generalization of a number of concepts in constraint programming (symmetry and substitutability), artificial intelligence (the pure literal rule in Boolean satisfiability) and operations research (dominance rules).
1 Introduction Significant work has been done over the past five years in the constraint programming community in the area of detection and exploitation of symmetry among search states. If symmetric states can be detected and avoided during search (typically through the posting of symmetry breaking constraints), significant gains in search efficiency can be made. Symmetry is quintessentially a bi-directional relationship: if state A is symmetric to state B, then state B is symmetric to A. In this paper we investigate the weakening of this bi-directionality. We call this relationship a pseudosymmetry. This term was introduced several years ago by Proll & Smith, who used the idea to improve performance on a template design problem [22]. However, it does not appear to have been adopted as a general modeling technique. We believe that pseudosymmetry is a powerful idea that can be applied to many asymmetric problems, and profitably combined with symmetry techniques on symmetric problems. The main contributions of this paper are: – the development of the idea of pseudosymmetry as a general technique for reducing search effort; – two empirical investigations showing that pseudosymmetry can be exploited to reduce search effort, leading to improved results for the Peaceable Armies of Queens problem;
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– the demonstration that pseudosymmetry generalizes a number of concepts that have existed separately for many years in the CP, AI, and OR communities — work done in the context of these more specialized notions immediately leads to promising research directions.
2 Pseudosymmetry Symmetry breaking exploits a property of solutions to a problem: that a solution exists if and only if symmetrically equivalent solutions exist. Constraints may be added to exclude all symmetric solutions but one, thus reducing the search space significantly. Symmetry breaking constraints have an important property: they are validity preserving, that is the original problem has a solution if and only if the new problem (with added constraints) has a solution. Moreover, all symmetrically equivalent solutions are recoverable from the reduced set of solutions. We weaken the notion of symmetry as follows. Assume we are able to identify a property, , of a solution (or partial solution) with the following attribute: if there exists a solution satisfying then there also exists at least one solution satisfying . This is a one-way relationship because the existence of a solution satisfying need not imply the existence of a solution satisfying . However, by only considering solutions satisfying we can reduce the search space without affecting validity. One way to do this, by analogy with symmetry breaking, is to add constraints to enforce . We call this technique pseudosymmetry breaking (throughout this paper we adopt the terminology of existing work on symmetry breaking). Pseudosymmetry breaking is justified by the following simple theorem: Theorem. Pseudosymmetry breaking is validity preserving. Proof. Consider two cases. (i) There is no solution satisfying . Then adding the constraint excludes no solutions. (ii) There is at least one solution satisfying . Then there also exists at least one solution satisfying , which the constraint does not exclude. By breaking pseudosymmetry we may lose access to the full set of solutions, unlike symmetry breaking (though in the problems used in this paper all solutions can be recovered). But this is unimportant for problems in which we are interested in finding any solution, any optimal solution (given an optimization problem), or in proving insolubility or optimality. As with symmetry, there are other ways to exploit pseudosymmetry besides posting constraints — notably by taking into account in a search heuristic so that only solutions with are searched for or by finding a new model of the problem that eliminates all states with property . Our empirical investigations focus on adding pseudosymmetry breaking constraints but we discuss other exploitation techniques in our review of previous work.
3 Empirical Investigations In many problems, symmetry breaking is based on geometric properties. For example in a chess problem such as N-queens, any solution can be rotated and/or reflected to obtain
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another solution. Crucially, the transformation is reversible. In this section, we investigate two problems, Maximum-Density Still Life and Peaceable Armies of Queens, and exploit geometric reasoning without the requirement of reversibility. 3.1 Maximum-Density Still Life As a first application of pseudosymmetry breaking, we consider the construction of a maximum density still-life (also called a stable pattern). We demonstrate that adding pseudosymmetry breaking constraints can result in reduction of search effort. The reduction obtained is marginal, but the problem serves as a proof-of-concept on a wellstudied problem in the symmetry breaking and OR literature. The Problem The game of Life was invented by Conway in the 1960s and popularized by Gardner [15]. In an infinite 2-dimensional array of cells, each cell is either alive or dead and has 8 neighbors that share 1 or 2 corners with it. The game is initialized by setting selected cells alive and the rest dead. Subsequently the array is transformed into a new pattern for as many iterations as desired using a few simple rules: – a cell with 2 living neighbors is unchanged in the new pattern; – a cell with 3 neighbors is alive in the new pattern; – any other cell is dead in the new pattern. A still-life is a pattern that does not change between iterations. A maximum density still-life is one with the greatest number of live cells, in a finite square region of cells (all cells outside are dead). An example is shown in Figure 1 with live cells represented by and dead cells by .
Fig. 1. A
maximum-density still-life
A Basic Model Two ILP formulations of this problem were given in [4] and one gave much better results with CPLEX. We use this second model, which has a 0/1 variable for each cell , 1 denoting a live cell and 0 a dead one. The constraints are as follows. Death by isolation (a cell with fewer than 2 live neighbors must be dead):
#%$ " !
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where denotes the variables corresponding to the neighbors of ’s cell. Death by overcrowding (a cell with more than 3 neighbors must be dead):
"! #
"! # #
Birth (a cell with 3 live neighbors must be alive):
where is a 3-element subset of from becoming alive:
. A boundary condition prevents cells outside
where , and are 3 cells lying in a line along the boundary of a density of at least :
. Finally, to ensure
To solve the problem we can solve a series of CSPs with increasing . Symmetry breaking can be used to improve the model but here we investigate only pseudosymmetries. Pseudosymmetry A pseudosymmetry occurs if we can find a pattern with the following property: it can be replaced by another pattern with no fewer live cells without affecting the still-life property. Such a pattern is shown in Figure 2: the four cells in the centre marked can be all live or all dead. To exclude the pattern in which the central 4 cells are dead:
The pattern may be partly off the edge of the finite region , as long as the four central cells are inside . We call these constraints . (A smaller pattern would be better but the rules of Life seem almost maliciously designed to prevent this; we invite the reader to find such patterns.)
" !# ! " !# ! Fig. 2. A still-life pattern for pseudosymmetry breaking
A second pseudosymmetry is based on the following observation: if there is a solution whose top row contains only dead cells then there is also a solution in which this is
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not true. We can simply translate the pattern one or more rows upward until a live cell appears in the top row. Therefore we can add a pseudosymmetry breaking constraint to exclude patterns in which the top row is all dead. Similarly, we can exclude patterns in which the left column is all dead:
We call these constraints . Even if there is no solution in which the top row is dead, the possibility may be explored by a search algorithm unless we add the constraints. Note that does not break a symmetry: patterns with dead top rows are not symmetrical with patterns containing at least one live top row cell. Results and Discussion We transform this basic model , with and without the and constraints, to linear pseudo-Boolean form and then apply a simple backtracker to solve the problem. Pseudo-Boolean form is a special form of 0/1 integer program that can be solved by SAT-based algorithms (see for example [1]). The backtracker uses a lexicographical variable ordering and a value ordering that tries 0 before 1. The variable ordering is overridden when a domain size becomes 1. The results are shown in Figures 3 and 4: “opt” means CPU time or number of backtracks needed to find an optimal solution (with a particular density) and “proof” means time or backtracks to prove that no denser solution exists (restarting the algorithm with a lower density bound equal to the known optimal value plus 1).
5 opt 999 proof 709 6 opt 7501 proof 25703 7 opt 161626 proof 159718 8 opt 4893631 proof 3289248
+ 877 639 3009 22292 145563 149054 4682075 3143639
+ + + 999 877 709 639 7484 2993 25700 22289 161542 145487 159686 149023 4882021 4672235 3282801 3137996
Fig. 3. Still-life results (backtracks)
In all cases the pseudosymmetry breaking constraints reduce or leave unchanged the required number of backtracks, demonstrating that pseudosymmetry can be used to prune a search space. Unfortunately the constraints result in little reduction in backtracks while creating CPU time overheads that outweigh their advantage. The constraints, on the other hand, are worth adding as they reduce both the backtracks and the CPU time. The results are comparable with the basic CP and IP results of [7] but not as good as their hybrid approach, the Bucket Elimination approach of [19] or the Dual Graph approach of [24]. Better results can almost certainly be obtained by
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5 opt 0.1s proof 0.1s 6 opt 0.5s proof 1.4s 7 opt 11s proof 11s 8 opt 6m2s proof 4m23s
+ + 0.1s 0.1s 0.1s 0.1s 0.2s 0.5s 1.2s 1.5s 10s 12s 11s 13s 5m48s 6m22s 4m18s 4m40s
+ + 0.1s 0.1s 0.2s 1.3s 10s 11s 6m34s 4m16s
Fig. 4. Still-life results (CPU time)
also exploiting symmetry breaking, adding implied constraints such as those of [7], and applying a hybrid CP/IP solver. However, we have shown that considerations of pseudosymmetry can lead to quite simple constraints that can be added to improve search performance even on a well-studied problem. We shall also reuse the idea of translational pseudosymmetry. 3.2 Peaceable Armies of Queens This chess problem was introduced by Bosch in the journal Optima [5]. While it has no practical applications, it has generated interest because even small instances are hard to solve. The problem is to place equally sized armies of black and white queens chess board so that no white queen can attack a black queen (or viceon an versa because attack is symmetrical), and to maximize the size of the armies. An integer program (IP) was defined by Plastria [6]. Smith et al. [23] defined and tested three constraint models for the problem, breaking symmetry by using SBDS (Symmetry Breaking During Search) [16]. As its name suggests, SBDS dynamically adds symmetry breaking constraints during search.
An Integer Model We define a new IP for the problem, drawing on ideas from the constraint models of [23]. Associate a pair of 0/1 variables and with each square. The take value 1 if there is a black queen on square and 0 otherwise while the are assigned to 1 if there is a white queen on square and 0 otherwise. In addition to these square variables, define 0/1 line variables and for each line (row, column or diagonal; one diagonal at each corner contains a single square). The optimization problem can be expressed as a series of CSPs with increasing lower bound on the number of black and white queens:
Any surplus queens may be removed to obtain a pattern with exactly black and white queens. The other constraints are as follows. No square or line may be both black and white:
#
#
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If a square is black [white] then its four associated line variables are black [white]:
#
#
where denotes the lines passing through square . If a line is black [white] then at least one of its associated squares is black [white]:
# !
where
# !
denotes the squares on line .
Symmetry Smith et al. use SBDS to break four rotational symmetries, two reflectional symmetries and one chromatic symmetry (all queen colors may be flipped to obtain another solution). Rather than using SBDS we add constraints to the model before search. This is probably weaker but it has the advantage that it can be used with any search algorithm. First the chromatic symmetry is broken by insisting that no white queens are placed on the top row:
$
If any queen is placed on the top row it must be black. For the rotational and reflectional symmetries, we consider the eight half-rows and half-columns at the edges of the board as binary representations of integers. We then post constraints to make the black top half of the left column represent the greatest number:
$ $ $ $ $ $ $
Pseudosymmetry Suppose that in a solution, queens are only placed in a smaller rect angle than , for example rows and columns . The solution can still be rotated and reflected within the smaller square, and also translated upward until a queen appears in the top row, and leftwards until a queen appears in the left column. The reverse is not true: a solution with queens in the top row and left column cannot necessarily be translated to remove them, because other queens might fall off the bottom row and right column. Thus we have two translational pseudosymmetries, which can be broken by insisting that at least one queen be placed in the top row and at least one queen be placed in the left column:
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where line denotes the left column. Combining these constraints with the the chromatic symmetry breaking constraint above, results in the constraint , requiring a black queen in the top row. Another type of pseudosymmetry occurs when placing a queen on a square cannot violate any constraint. There are three cases. Firstly, if a solution exists with an empty square whose lines are uncolored then a solution also exists with a queen of either color in that square (recall that in our formulation only lower bounds are placed on the sizes of the armies). We call this the uncolored pseudosymmetry and it can be broken by placing a queen in that square; more specifically, a black queen. Secondly, if a solution exists in which an empty square has no white lines and at least one black line passing through it, then a black queen may be placed there. We call this the black pseudosymmetry. Thirdly, if a solution exists in which an empty square has no black lines and at least one white line, then a white queen may be placed there. This is the white pseudosymmetry. Two families of constraints can be used to break these pseudosymmetries. One breaks both the black and uncolored:
and the other breaks the white:
Results and Discussion As with the still-life models, these models are transformed to pseudo-Boolean form. We apply a naive backtracker using a lexicographical variable ordering: first the square variables row by row, then the line variables. We also tried the much more sophisticated PBS solver [1] but surprisingly it gave inferior results, though on other problems it is far better. It seems that a static variable ordering is best on this particular problem. Figure 5 shows the CPU times taken for the algorithm to find and prove optimal solutions for various values of . It shows results for the basic model (basic), the basic model with symmetry breaking (symm), with pseudosymmetry breaking (pseu), with both (both), and both with a modified (but still static) variable ordering heuristic (+VO). The modified variable ordering chooses variables with the highest score where and are the number of positive and negative occurrences of the variable, breaking ties lexicographically. The figure also shows results for ILOG Solver and Eclipse with and without SBDS on a basic squares-only model (basic and symm), and ILOG Solver results with SBDS on the best (unattacked squares) model and a most-unattacked-square variable ordering heuristic (U+VO). The results with Solver and Eclipse are taken from Smith et al. [23]. denotes the number of queens. The machine for the pseudo-Boolean solving is a 733 MHz Pentium 3, whereas Smith et al. use a 1.6 GHz Pentium 4 for Eclipse and a 600 MHz Celeron PC for ILOG Solver.
instance on a CPLEX on Plastria’s IP model took 4 hours to solve the 200 MHz Pentium PC, but all the constraint models give shorter times. Both symmetry and pseudosymmetry breaking make a significant difference to search time. Though
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4 5 6 7 8 9 10 11 12 13
2 4 5 7 9 12 14 17 21 24
pseudo-Boolean basic symm pseu both 0.1 0.01 0.1 0.1 0.12 0.02 0.08 0.01 1.6 0.15 0.6 0.07 29 2.2 6 0.49 236 41 93 5.4 26091 1704 1262 93 879 12379
+VO 0.01 0.02 0.08 0.66 5.25 31 298 2874 17868 247331
Eclipse basic symm 0.01 0.04 0.13 0.16 3.2 1.7 78 27 3500 640
ILOG Solver basic symm U+VO 0.03 0.02 0.02 0.11 0.04 0.03 2.9 0.41 0.13 56 7.8 1.12 2100 240 11.7 116 2460 37100
Fig. 5. Results for several algorithms on the Peaceable Armies of Queens problem
our symmetry breaking approach is simple it achieves speedups comparable to those of SBDS on this problem. Pseudosymmetry breaking achieves similar speedups, and the speedup appears to increase with until it gives better results than symmetry breaking. Applying both symmetry and pseudosymmetry breaking gives even better results than ILOG Solver, and adding the modified variable ordering further reduces execution times so that we are able to increase by 2. These results show the potential benefits of pseudosymmetry breaking, and that it can profitably be combined with symmetry breaking. Figure 6 shows the first known optimal solutions, as far as we are aware, for
and
. The solution for
actually contains 25 white queens and 24 black ones; to obtain a solution with equal-sized armies we can simply remove any white queen.
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b b b . . . . . . . . .
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. . . . . . . . w w w w
Fig. 6. Optimal
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4 Perspectives on Related Work We have already mentioned prior work of Proll & Smith [22] in which the term pseudosymmetry was introduced. We now survey other work which we believe is closely related to pseudosymmetry. By making these connections explicit we hope to stimulate further cross-fertilization between different fields of computer science. 4.1 Symmetry and Substitutability Probably the most popular approach to exploiting symmetry is to add constraints to the problem formulation, so that each equivalence class of solutions to the original problem corresponds to a single solution in the new problem. Another possibility is to modify the search algorithm to detect and exploit symmetries dynamically [2, 9, 12, 16]. In particular, conditional symmetry [17] handles cases which which solutions become symmetric under certain conditions during search. A related idea for planning problems is dynamic symmetry [13]. It is not required that pseudosymmetric solutions ever become interchangeable, therefore pseudosymmetry is more general than conditional and dynamic symmetry. For example, consider the translational pseudosymmetries in the Still Life and Armies of Queens problems. To establish a conditional symmetry we would need to establish conditions under which solutions with and without empty top rows become interchangeable. To do this we need constraints involving most or all of the problem variables, to ensure that the cost constraint does not become violated. In contrast, pseudosymmetry requires only constraints on the top row variables. By dropping the requirement of interchangeability we can prune the search space under weaker conditions. Freuder defined the concept of substitutability for CSPs [14]. A domain value for a variable is substitutable for another value of if and only if any solution con taining can be transformed into another solution by reassigning . Freuder also defined interchangeability which corresponds to two-way substitutability. Quite a literature has developed around interchangeability and substitutability, which we do not review here for reasons of space. As can be seen from the definitions, substitutability bears the same relation to interchangeability as pseudosymmetry does to symmetry. Furthermore, Benhamou [3] showed that interchangeability is a special type of symmetry, and we therefore claim that substitutability is a special form of pseudosymmetry. 4.2 Pure Literals in SAT In a SAT problem, it may be the case that a variable only occurs positively as but not negatively as (or vice-versa). This can be exploited to reduce search effort by adding a unary clause (or ). For example given a SAT problem
literal appears but does not, so we can add the unary clause to the problem. This makes the first clause redundant so it can be deleted, and variable removed from the problem:
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There are no further pure literals. But if during search we set to false then the second clause becomes true and can be deleted, leaving the problem
Now literal is pure, thus the pure literal rule can be applied during search as well as during preprocessing. The rule was first proposed in 1960 [8] and is still used in some SAT solvers. It can be viewed as an application of pseudosymmetry as follows. Given a SAT problem in which variable appears only positively, suppose there is a solution to the SAT problem in which is false. Then there is also a solution in which is true, because negating from false to true can not cause a clause containing the literal to become false. Here the property is a is false, and we eliminate such solutions by adding the unary clause . 4.3 Dominance Rules Dominance rules or dominance relations are often exploited in combinatorial search for optimization problems within operations research and, indeed, AI [18]. It can be shown that the basic form of a dominance rule is identical to pseudosymmetry: some property is identified such that a solution with property implies a solution with of equal or higher quality. Therefore, only solutions with need be considered. Typically, this means that the heuristics used in branching as well as the bounding conditions are designed with in mind. In contrast with symmetry breaking constraints, often there is no explicit representation of the dominance rule but rather it is built into the logic of the search. For example, in job shop scheduling (JSP) dominance rules are used to define a subset of problems states that are searched over. A solution to a JSP can be represented as a complete ordering of activities on each resource that respects the temporal constraints that exist between activities. A schedule is semi-active if all activities are maximally left-shifted such that no activity can be moved further left without changing the activity sequence. For regular optimization criteria, such as makespan minimization, if there exists an optimal solution that is not semi-active, then there exists a corresponding optimal semi-active solution. The semi-active solution can be found by simply shifting all activities to the left without changing the sequences. Such shifting cannot increase the makespan, therefore if the non-semi-active solution was optimal, so is the semiactive solution. Figure 7(A) and (B) present a non-semi-active solution and the corresponding semi-active solution respectively (assuming precedence relations among the activities in the same jobs). Activities J0A1 and J1A2 can both be shifted to the left without changing the activities sequences or violating any constraints. The semi-active rule is commonly exploited in local search techniques for scheduling where many of the current state-of-the-art neighborhoods for JSP are defined to traverse only semi-active solutions [25, 21]. Another dominance rule limiting the search to active solutions can also be applied to JSPs. An active solution is a solution in which no activities can be executed earlier without delaying the execution of another activity. In semi-active solutions while no activities can be shifted left, it might be possible to “jump” an activity over other activities
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J2A1 $# $# $# $# J1A1 "! "! "! J0A2 "!! "!! "" J2A0 J1A0J2A2$# $# $# &%% $# &%% J0A1 &%% &%% "! &%% "! &%% "! J1A2 &&&&&&
R0 R1 R2
R0 R1 R2
J2A1 J1A1 J0A2 J2A0 J1A0J2A2 J0A1 J1A2
(B) Semi−active J2A1 J0A2 J1A1 R0 J2A0 J1A0J2A2J0A1 R1 J1A2 R2
J0A0
J0A0
(A) Non−semi−active
J0A0
(C) Active
Fig. 7. An example of (A) a non-semi-active schedule (i.e., J0A1 and J1A2 can be left-shifted), (B) a semi-active but not active schedule (i.e., J0A0 can be moved earlier without disruption of other activities), and (C) an active schedule. The precedence constraints within jobs are implicit: J A0 must end before the start of J A1 which much end before the start of J A2.
'
'
'
in order to insert it into empty space earlier in the schedule. The insertion must be done however without pushing any activities to the right or, of course, without violating any temporal constraints. Figure 7(C) displays an active solution. Comparing it with Figure 7(B), we can see that activity J0A0 has been jumped over J2A2 to take advantage of a large empty space on resource R2, allowing J0A1 to be left-shifted. Any optimal non-active solution can be transformed into an optimal active solution by moving activities to the left without delaying other activities. Since no other activities are delayed, the makespan cannot be increased and therefore the active schedule is optimal. The active property is also often exploited in searching for solutions. For example, the “schedule-or-postpone” heuristic [20] is a constructive heuristics that generates only active solutions. At a search node all “selectable” activities are identified, and one is heuristically chosen and scheduled at its earliest start time. On backtracking, the activity is postponed, meaning that it is not selectable until its earliest start time has been changed by constraint propagation. If there are no selectable activities, search backtracks to the most recent heuristic decision. If the selectable activities are all those which start before the earliest end time of all unscheduled activities, which are not postponed, and which have all their upstream activities already scheduled, it can be shown that only (and all) the active schedules will be enumerated [10]. 4.4 Problem Reformulation By generalizing techniques of symmetry exploitation and dominance rules, we have seen two ways of exploiting pseudosymmetry: the addition of pseudosymmetry breaking constraints and the incorporation of pseudosymmetries into search heuristics. A third possibility is to reformulate a problem more drastically so that the search states we wish to exclude are not modeled. For the Peaceable Armies of Queens problem the best constraint model tested by Smith et al. [23] was the unattacked queens model. That model represents only the white queens, ensuring that there are at least as many unattacked squares as white queens; black queens are implicitly placed in these squares. The stated advantages of this model are that the search variables have smaller domains and that there are fewer constraints.
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Another perspective on this model is that by merging the cases of a square containing no queen and a black queen, it implicitly breaks the black and uncolored (but not the white or translational) pseudosymmetries. Therefore, perhaps part of the advantage of this model can be traced to the absence of these pseudosymmetries. Our point is that, in retrospect, the unattacked queens model could have been inspired by the identification and elimination of pseudosymmetries. Given that finding good constraint models is a poorly understood and difficult process, heuristics that can be applied (initially by humans) to create good models are an important research direction. We suggest that the identification and elimination of pseudosymmetries can be one such model-building heuristic.
5 Conclusion Building on the work of Proll & Smith [22], we have demonstrated that pseudosymmetry is a general technique for reduction of search effort. This was shown explicitly in two empirical studies resulting, in one case, in improved results over previous studies. We have shown that pseudosymmetry is a generalization of concepts that exist in several fields that deal with combinatorial search. Besides being a natural extension of recent work on symmetry in constraint programming it is closely related to the use of dominance rules in operations research (but need not be based on cost reasoning), and to techniques used in propositional satisfiability. Unifying the terminology and emphasizing the generality of pseudosymmetry may help the modeler to notice further opportunities for its application. We have found that thinking along these lines can lead to new constraints that can be used to further prune problems, and we hope to stimulate further research on pseudosymmetry. Acknowledgments Thanks to Brahim Hnich and Barbara Smith for helpful discussions on symmetry and pseudosymmetry.
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