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## Efficient Constraint Propagation Engines

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Citations: | 61 - 9 self |

### Citations

1328 | S.: Chaff: Engineering an efficient SAT solver
- Moskewicz, Madigan, et al.
- 2001
(Show Context)
Citation Context ...ning of the propagator f to handle what happens when moving from D to D ′ . Fully dynamic event sets are closely related to the watched literals approach to improving unit propagation in SAT solving [=-=Moskewicz et al. 2001-=-]. A clause in CNF corresponds to the constraint �n i=1 xi ≥ 1 explored in Example 5.16. Using watched literals, unit propagation only considers a clause for propagation if one of two watched literals... |

1138 |
Consistency in networks of relations
- Mackworth
- 1977
(Show Context)
Citation Context ...Many propagators in practice are designed to maintain some form of consistency: usually domain or bounds. But note that many more do not. The most successful consistency technique is arc consistency [=-=Mackworth 1977-=-], which ensures that for each binary constraint, every value in the domain of the first variable, has a supporting value in the domain of the second variable that satisfied the constraint. Arc consis... |

405 |
Programming with Constraints: An Introduction
- Marriott, Stuckey
- 1998
(Show Context)
Citation Context ...has been called generalized arc consistency [Mohr and Masini 1988], as well as domain consistency [Van Hentenryck et al. 1991; 1998] (which is the terminology we will use), and hyper-arc consistency [=-=Marriott and Stuckey 1998-=-]. A domain D is domain consistent for a constraint c if D is the least domain containing all solutions θ ∈ D of c, that is, there does not exist D ′ ⊏ D such that θ ∈ D ∧ θ ∈ c → θ ∈ D ′ . Define the... |

375 |
A filtering algorithm for constraints of difference in CSPs
- Régin
- 1994
(Show Context)
Citation Context ...ed by fL(D)(x1) = D(x1) ∩ � 2⌈1 2 infD x1⌉ .. 2⌊1 2 supD(x1)⌋ � fL(D)(x) = D(x) x �= x1 might be given priority unary, while fE and fF might be given priority binary. The domain propagator defined in =-=[29]-=- for the alldifferent constraint ∧ n i=1 ∧n j=i+1 xi �= xj (with complexity O(n 2.5 )) might be given priorityquadratic. The alldifferent bounds(Z) propagator defined in [28] (with complexity O(n log ... |

256 | Principles of Constraint Programming
- Apt
- 2003
(Show Context)
Citation Context ...D) iter(F, D) = ⊓ f∈F f(D) where gfp denotes the greatest fixpoint w.r.t ⊑ lifted to functions. Note that by inverting the direction of ⊑ we could equally well phrase this as a least fix point (as in =-=[1]-=-). But the current presentation emphasizes the reduction of domains as computation progresses. Domain and bounds propagators A consistency notion C gives a condition on domains with respect to constra... |

250 |
The OPL Optimization Programming Language
- Hentenryck
- 1999
(Show Context)
Citation Context ...re-7) computes a magic square of size 5 × 5 (7 × 7) using linear equation propagators and a single alldifferent propagator (prob019 in [10]). • warehouse solves a warehouse location problem following =-=[32]-=-. B Evaluation Platform All experiments use Gecode, a C ++-based constraint programming library [12]. Gecode is one of the fastest constraint programming systems currently available, benchmarks compar... |

233 |
T.C.: Arc and path consistency revisited
- Mohr, Henderson
- 1986
(Show Context)
Citation Context ...nd ran are useful for tracking endpoint-relevance and rangeequivalence [Schulte and Stuckey 2005]. The neq event has been used in e.g. Choco [Laburthe 2000] and B-Prolog [Zhou 2006] for building AC4 [=-=Mohr and Henderson 1986-=-] style propagators. Example 5.2 Events. Let D(x1) = {1, 2, 3}, D(x2) = {3, 4, 5, 6}, D(x3) = {0, 1}, and D(x4) = {7, 8, 10} while D ′ (x1) = {1, 2}, D ′ (x2) = {3, 5, 6}, D ′ (x3) = {1} and D ′ (x4) ... |

193 | An open-ended finite domain constraint solver
- Carlsson, Ottosson, et al.
- 1997
(Show Context)
Citation Context ...y use sup D x then ubc(x) is in the event set. If they use the entire domain D(x) then dmc(x) is in the event set. ✷ Indexical propagation solvers [Van Hentenryck et al. 1998; Codognet and Diaz 1996; =-=Carlsson et al. 1997-=-] are based on such reasoning. They define propagators in the form f(D)(x) = D(x) ∩ e(D) where e is an indexical expression. The event ACM Transactions on Programming Languages and Systems, Vol. V, No... |

186 | Y.: Design, implementation, and evaluation of the constraint language cc(fd
- Hentenryck, Saraswat, et al.
- 1994
(Show Context)
Citation Context ...ied the constraint. Arc consistency can be naturally extended to constraints of more than two variables. This extension has been called generalized arc consistency [24], as well as domain consistency =-=[33, 34]-=- (which is the terminology we will use), and hyper-arc consistency [22]. A domain D is domain consistent for a constraint c if D is the least domain containing all solutions θ ∈ D of c, that is, there... |

170 | Compiling constraints in clp(fd
- Codognet, Diaz
- 1996
(Show Context)
Citation Context ...nt set, similarly if they use sup D x then ubc(x) is in the event set. If they use the entire domain D(x) then dmc(x) is in the event set. ✷ Indexical propagation solvers [Van Hentenryck et al. 1998; =-=Codognet and Diaz 1996-=-; Carlsson et al. 1997] are based on such reasoning. They define propagators in the form f(D)(x) = D(x) ∩ e(D) where e is an indexical expression. The event ACM Transactions on Programming Languages a... |

123 |
A regular language membership constraint for finite sequences of variables
- Pesant
- 2004
(Show Context)
Citation Context ...3, 4, 5}. This is not a fixpoint for fG because of the hole (that is 1, 2, 4 ∈ D(x1)) in the original domain of x1. ✷ Example 4.7 [Multiple variable occurrences] The regular constraint introduced in =-=[27]-=- constrains a sequence of variables to take values described by a regular expression (or a corresponding finite automaton). A common case for the regular constraint is to express cyclic patterns by pe... |

119 | ECLiPSe: A Platform for Constraint Logic Programming,
- Wallace
- 1997
(Show Context)
Citation Context ...s Laboratory 2004], Mozart [Mozart Consortium 1999]). The two levels are often not entirely based on cost: in SICStus all indexicals have high priority and all other lower priority. While ECL i PS e [=-=Wallace et al. 1997-=-; Harvey 2004] supports twelve priority levels, its finite domain solver also uses only two priority levels where another level is used to support constraint debugging. A similar, but more powerful ap... |

109 |
Good old discrete relaxation
- Mohr, Masini
- 1988
(Show Context)
Citation Context ... of the second variable that satisfied the constraint. Arc consistency can be naturally extended to constraints of more than two variables. This extension has been called generalized arc consistency [=-=Mohr and Masini 1988-=-], as well as domain consistency [Van Hentenryck et al. 1991; 1998] (which is the terminology we will use), and hyper-arc consistency [Marriott and Stuckey 1998]. A domain D is domain consistent for a... |

79 | Constraint satisfaction using constraint logic programming - Hentenryck, Simonis, et al. - 1992 |

72 |
A fast algorithm for the bound consistency of alldiff constraints
- Puget
- 1998
(Show Context)
Citation Context ...ator defined in [Régin 1994] for the alldifferent constraint ∧ n i=1 ∧n j=i+1 xi �= xj (with complexity O(n 2.5 )) might be given priority quadratic. The alldifferent bounds(Z) propagator defined in [=-=Puget 1998-=-] (with complexity O(n log n)) might be given priority linear. ✷ Priorities in effect force many more fixpoints to be calculated. A fixpoint of all propagators at priority level i and lower must be re... |

70 | Constraint processing in cc(fd
- Hentenryck, Saraswat, et al.
- 1992
(Show Context)
Citation Context ...ied the constraint. Arc consistency can be naturally extended to constraints of more than two variables. This extension has been called generalized arc consistency [24], as well as domain consistency =-=[33, 34]-=- (which is the terminology we will use), and hyper-arc consistency [22]. A domain D is domain consistent for a constraint c if D is the least domain containing all solutions θ ∈ D of c, that is, there... |

64 | W.: Constraint-Based Scheduling - Baptiste, Pape, et al. - 2003 |

50 | Heterogeneous constraint solving
- Benhamou
- 1996
(Show Context)
Citation Context ...′ in F − Q is at fixpoint at the start of the loop. The incremental propagation solver isolv with this definition of new (assuming Fo = ∅) is more or less equivalent to the propagation algorithms in [=-=Benhamou 1996-=-] and [Apt 2003, page 267]. Example 3.2 Incremental propagation. Consider the problem with constraints ACM Transactions on Programming Languages and Systems, Vol. V, No. N, Month 20YY.s8 · C. Schulte ... |

46 |
Gecode: A generic constraint development environment, 2006. Available at http: //www.gecode.org
- Team
(Show Context)
Citation Context ...propagation. This framework is realized in finite domain constraint programming systems, such as SICStus Prolog [Intelligent Systems Laboratory 2004], ILOG Solver [ILOG S.A. 2000], and Gecode [Gecode =-=Team 2006-=-] that have been successfully applied to many real-life industrial applications. At the core of a finite domain constraint programming system is a constraint propagation engine that repeatedly execute... |

44 |
Choco: Implementing a CP kernel
- Laburthe
- 2000
(Show Context)
Citation Context ...hanges the domain is likely to be executed again to check it is at fixpoint. Almost all constraint propagation solvers take into account static fixpoint reasoning (for example ILOG Solver [17], Choco =-=[19]-=-, SICStus [18], and Gecode [12]). Some systems even only allow idempotent propagators (for example Mozart [26]). 4.2 Dynamic Fixpoint Reasoning Even if a propagator is not idempotent we can often dete... |

41 | I (2006) Watched literals for constraint propagation in Minion
- IP, Jefferson, et al.
(Show Context)
Citation Context ...ly considers a clause for propagation if one of two watched literals in the clause becomes false. Recently, the idea of watched literals has been used in constraint programming for the Minion solver [=-=Gent et al. 2006-=-]. Watched literals differ from the events we concentrate on here since they take into account values (similar to the neq(x, a) event) and require the knowledge of which variable is changed by propaga... |

31 | Improving linear constraint propagation by changing constraint representation
- Harvey, Stuckey
(Show Context)
Citation Context ... .. 4]. Hence fF(fF(D)) = D ′′ = D ′ = fF(D). ✷ We can always create an idempotent propagator f ′ from a propagator f by defining f ′ (D) = solv({f}, D). Indeed, in some implementations (for example =-=[16]-=-) prop(3x1 = 2x2) is defined as the fixpoint of applying fF. Assume that idem(f) = {f} if f is an idempotent propagator and idem(f) = ∅ otherwise. The definition of new is improved by taking idempoten... |

28 | Finite-domain constraint propagators in action rules. Theory and Practice of Logic Programming
- Zhou
- 2005
(Show Context)
Citation Context ... the value d, that is d ∈ D(x) and d �∈ D ′ (x) The events two and ran are useful for tracking endpoint-relevance and rangeequivalence [31]. The neq event has been used in e.g. Choco [19] and BProlog =-=[37]-=- for building AC4 [23] style propagators. Example 5.2 [Events] Let D(x1) = {1, 2, 3}, D(x2) = {3, 4, 5, 6}, D(x3) = {0, 1}, and D(x4) = {7, 8, 10} while D ′ (x1) = {1, 2}, D ′ (x2) = {3, 5, 6}, D ′ (x... |

27 | P.J.: Speeding up constraint propagation - Schulte, Stuckey - 2004 |

27 | When do bounds and domain propagation lead to the same search space - Schulte, Stuckey - 2005 |

20 | Finite domain bounds consistency revisited
- Choi, Harvey, et al.
- 2006
(Show Context)
Citation Context ...bounds consistency is to relax the consistency requirement to apply only to the lower and upper bounds of the domain of each variable x. There are a number of different notions of bounds consistency [=-=Choi et al. 2004-=-], we give the two most common here. A domain D is bounds(Z) consistent for a constraint c, vars(c) = {x1, . . . , xn}, if for each variable xi, 1 ≤ i ≤ n and for each di ∈ {infD xi, sup D xi} there e... |

20 | Dynamic optimization of interval narrowing algorithms. Journal of Logic Programming Forthcoming
- Lhomme, Gotlieb, et al.
- 1998
(Show Context)
Citation Context ...FIFO traversal. Prioritizing particular operations during constraint propagation is important in general. For interval narrowing, prioritizing constraints can avoid slow convergence, see for example [=-=Lhomme et al. 1998-=-]. The prioritizing of propagators by cost is important, inverting the priorities can lead to significant disadvantages. Example 6.3 Inverted priorities. Consider executing Example 6.2 with inverted p... |

20 |
The Mozart programming system
- Consortium
- 1999
(Show Context)
Citation Context ... solvers take into account static fixpoint reasoning (for example ILOG Solver [17], Choco [19], SICStus [18], and Gecode [12]). Some systems even only allow idempotent propagators (for example Mozart =-=[26]-=-). 4.2 Dynamic Fixpoint Reasoning Even if a propagator is not idempotent we can often determine that f(D) is a fixpoint of f for a specific domain D. For simplicity we assume a function fix(f, D) that... |

15 | Revisiting the lexicographic ordering constraint - Carlsson, Beldiceanu - 2002 |

14 | Advisors for incremental propagation
- Lagerkvist, Schulte
- 2007
(Show Context)
Citation Context ...ation by maintaining the set of propagators Q. Information about which variables change and how their domain changes during propagation as needed for watched literals requires additional effort, see [=-=Lagerkvist and Schulte 2007-=-]. 5.4 Dynamic Event Sets Experiments Table VIII shows the comparison of monotonic and fully dynamic event sets to a propagation solver using static event sets with {fix, bc, dmc} events and prioritie... |

12 |
Amusements in Mathematics
- Dudeney
- 1917
(Show Context)
Citation Context ...cing problem from [Van Hentenryck et al. 1992] using element, exactly, and linear equation propagators (prob001 in [CSPLib 2006]). — crowded-chess-7 places different chess pieces on a 7×7 chessboard [=-=Dudeney 1958-=-]. It uses exactly, element, domain consistent alldifferent, and bounds(R) consistent linear equation propagators. ACM Transactions on Programming Languages and Systems, Vol. V, No. N, Month 20YY.s38 ... |

11 |
Mehmet Dincbas. Constraint satisfaction using constraint logic programming
- Hentenryck, Simonis
- 1992
(Show Context)
Citation Context ...ue of x2 will never cause propagation. A minimal definition of es(fK, D) is {lbc(x0), lbc(x1), ubc(x1)}. ✷ Example 5.12 [exactly propagator] Another example is a propagator for the exactly constraint =-=[35]-=-: exactly([x1, . . . , xn], m, k) states that exactly m out of the variables x1, . . . , xn are equal to a value k. As soon as one of the xi becomes different from k, all events for xi can be ignored.... |

5 | Implementing constraint propagation by composition of reductions - Granvilliers, Monfroy - 2003 |

4 | Proceedings of TRICS: Techniques foR Implementing Constraint programming Systems, a postconference workshop of CP 2000 - Beldiceanu, Harvey, et al. - 2000 |

4 | Using weaker functions for constraint propagation over real numbers - Monfroy - 1999 |

4 |
Nuijten: Constraint-Based Scheduling
- Baptiste, Pape, et al.
- 2003
(Show Context)
Citation Context ...ferent constraint. Another area where staged propagators can be used is constraint-based scheduling, where typically different propagation methods with different strength and efficiency are available =-=[2]-=-. Staging is not limited to expensive propagators, it is already useful for binary (for example, combining bounds and domain propagation for the absolute value constraint abs(x) = y) and ternary const... |

4 |
ILOG Solver 5.0: Reference Manual
- ILOG
- 2000
(Show Context)
Citation Context ...ach to solving CSPs combines backtracking tree search with constraint propagation. This framework is realized in finite domain constraint programming systems, such as SICStus Prolog [18], ILOG Solver =-=[17]-=-, and Gecode [12] that have been successfully applied to many real-life industrial applications. At the core of a finite domain constraint programming system is a constraint propagation engine that re... |

3 | Constraint reduction at the type level
- Savéant
(Show Context)
Citation Context ... fix(f, D) evset[f] := es(f, D ′ ) return F ′ Every time a propagator f is applied its event set is updated to take into account newly available information. A related idea is the “type reduction” of =-=[30]-=- where propagators are improved as more knowledge on domains (here called types) becomes available. For example, the implementation of x0 = x1 × x2 will be replaced by a more efficient one, when all e... |

2 |
CSPLib: a problem library for constraints, 2006. Available from http://www.csplib.org
- CSPLib
(Show Context)
Citation Context ...s (including linear constraints) and naive propagation for alldifferent. • all-interval-500 computes a series of numbers where the distances between adjacent numbers are pairwise distinct (prob007 in =-=[10]-=-). The model uses a single bounds(Z) consistent alldifferent propagator and many binary absolute value (abs(x) = y) and ternary minus propagators. 39sTable 14: Example characteristics. Example variabl... |

1 |
CHOCO: implementing a CP kernel. See Beldiceanu et al
- Laburthe
- 2000
(Show Context)
Citation Context ...main is likely to be executed again to check it is at fixpoint. Almost all constraint propagation solvers take into account static fixpoint reasoning (for example ILOG Solver [ILOG S.A. 2000], Choco [=-=Laburthe 2000-=-], SICStus [Intelligent Systems Laboratory 2004], and Gecode [Gecode Team 2006]). Some systems even only allow idempotent propagators (for example Mozart [Mozart Consortium 1999]). Example 4.4 Idempot... |

1 |
The Mozart programming system. Available from www.mozart-oz.org
- Consortium
- 1999
(Show Context)
Citation Context ...[ILOG S.A. 2000], Choco [Laburthe 2000], SICStus [Intelligent Systems Laboratory 2004], and Gecode [Gecode Team 2006]). Some systems even only allow idempotent propagators (for example Mozart [Mozart =-=Consortium 1999-=-]). Example 4.4 Idempotent incremental propagation. Consider the problem from Example 3.2 where idempotent propagators f ∗ c (D) = solv({fc}, D) and f ∗ d (D) = solv({fd}, D) are used. Starting from t... |

1 |
Efficient Constraint Propagation Engines · 41
- Régin
- 1994
(Show Context)
Citation Context ... and Systems, Vol. V, No. N, Month 20YY.sEfficient Constraint Propagation Engines · 25 might be given priority unary, while fe and ff might be given priority binary. The domain propagator defined in [=-=Régin 1994-=-] for the alldifferent constraint ∧ n i=1 ∧n j=i+1 xi �= xj (with complexity O(n 2.5 )) might be given priority quadratic. The alldifferent bounds(Z) propagator defined in [Puget 1998] (with complexit... |

1 |
Constraint reduction at the type level. See Beldiceanu et al
- Savéant
- 2000
(Show Context)
Citation Context ...fix(f, D) evset[f] := es(f, D ′ ) return F ′ Every time a propagator f is applied its event set is updated to take into account newly available information. A related idea is the “type reduction” of [=-=Savéant 2000-=-] where propagators are improved as more knowledge on domains (here called types) becomes available. For example, the implementation of x0 = x1 × x2 will be replaced by a more efficient one, when all ... |

1 | foR Implementing Constraint programming Systems, a post-conference workshop of CP 2000 - Beldiceanu, Harvey, et al. - 2000 |

1 |
Dominique-Benoit Guinaudeau, and Andre Guillard. CHIC lessons on CLP methodology
- Chamard, Fischler
- 1995
(Show Context)
Citation Context ...t. When using a variant of thelex propagator proposed in [5], events can be removed as soon as the order among pairs of variables being compared can be decided. For the generalized element constraint =-=[7]-=- where the array elements are variables, events for a variable from the array can be safely removed as soon as the variable becomes known to be different from the result of the element constraint. Usi... |

1 |
Eric Monfroy. Implementing constraint propagation by composition of reductions
- Granvilliers
- 2003
(Show Context)
Citation Context ...he priorities into fewer levels, for example into three levels unary-ternary, linear-quadratic,cubic-veryslow. Another model for priorities in constraint propagation based on composition operators is =-=[14]-=-. The model, however, runs all propagators of lower priority before switching propagation back to propagators of higher priority. This model does not preempt computing a fixpoint for a low priority. T... |