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Data structures for generalised arc consistency for extensional constraints
 In Proceedings of the Twenty Second Conference on Artificial Intelligence
, 2007
"... Extensional (table) constraints are an important tool for attacking combinatorial problems with constraint programming. Recently there has been renewed interest in fast propagation algorithms for these constraints. We describe the use of two alternative data structures for maintaining generalised ar ..."
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Cited by 21 (8 self)
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Extensional (table) constraints are an important tool for attacking combinatorial problems with constraint programming. Recently there has been renewed interest in fast propagation algorithms for these constraints. We describe the use of two alternative data structures for maintaining generalised arc consistency on extensional constraints. The first, the NextDifference list, is novel and has been developed with this application in mind. The second, the trie, is well known but its use in this context is novel. Empirical analyses demonstrate the efficiency of the resulting approaches, both in GACschema, and in the watchedliteral table constraint in Minion.
P.: A Constraint Store Based on Multivalued Decision Diagrams
 Principles and Practice of Constraint Programming (CP 2007). Lecture Notes in Computer Science
, 2007
"... Abstract. The typical constraint store transmits a limited amount of information because it consists only of variable domains. We propose a richer constraint store in the form of a limitedwidth multivalued decision diagram (MDD). It reduces to a traditional domain store when the maximum width is on ..."
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Cited by 19 (9 self)
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Abstract. The typical constraint store transmits a limited amount of information because it consists only of variable domains. We propose a richer constraint store in the form of a limitedwidth multivalued decision diagram (MDD). It reduces to a traditional domain store when the maximum width is one but allows greater pruning of the search tree for larger widths. MDD propagation algorithms can be developed to exploit the structure of particular constraints, much as is done for domain filtering algorithms. We propose specialized propagation algorithms for alldiff and inequality constraints. Preliminary experiments show that MDD propagation solves multiple alldiff problems an order of magnitude more rapidly than traditional domain propagation. It also significantly reduces the search tree for inequality problems, but additional research is needed to reduce the computation time. 1
Optimization of simple tabular reduction for table constraints
 In Proceedings of CP’08
, 2008
"... Abstract. Table constraints play an important role within constraint programming. Recently, many schemes or algorithms have been proposed to propagate table constraints or/and to compress their representation. We show that simple tabular reduction (STR), a technique proposed by J. Ullmann to dynamic ..."
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Cited by 16 (9 self)
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Abstract. Table constraints play an important role within constraint programming. Recently, many schemes or algorithms have been proposed to propagate table constraints or/and to compress their representation. We show that simple tabular reduction (STR), a technique proposed by J. Ullmann to dynamically maintain the tables of supports, is very often the most efficient practical approach to enforce generalized arc consistency within MAC. We also describe an optimization of STR which allows limiting the number of operations related to validity checking or search of supports. Interestingly enough, this optimization makes STR potentially r times faster where r is the arity of the constraint(s). The results of an extensive experimentation that we have conducted with respect to random and structured instances indicate that the optimized algorithm we propose is usually around twice as fast as the original STR and can be up to one order of magnitude faster than previous stateoftheart algorithms on some series of instances. 1
Maintaining Generalized Arc Consistency on Ad Hoc rary Constraints
"... In many reallife problems, constraints are explicitly defined as a set of solutions. This ad hoc (table) representation uses exponential memory and makes support checking (for enforcing GAC) difficult. In this paper, we address both problems simultaneously by representing an ad hoc constraint with ..."
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Cited by 16 (0 self)
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In many reallife problems, constraints are explicitly defined as a set of solutions. This ad hoc (table) representation uses exponential memory and makes support checking (for enforcing GAC) difficult. In this paper, we address both problems simultaneously by representing an ad hoc constraint with a multivalued decision diagram (MDD), a memory efficient data structure that supports fast support search. We explain how to convert a table constraint into an MDD constraint and how to maintain GAC on the MDD constraint. Thanks to a sparse set data structure, our MDDbased GAC algorithm, mddc, achieves full incrementality in constant time. Our experiments on structured problems, car sequencing and stilllife, show that mddc is a fast GAC algorithm for ad hoc constraints. It can replace a Boolean sequence constraint [1], and scales up well for structural MDD constraints with 208 variables and 340984 nodes. We also show why it is possible for mddc to be faster than the stateoftheart generic GAC algorithms in [2–4]. Its efficiency on nonstructural ad hoc constraints is justified empirically.
Efficient Reasoning for Nogoods in Constraint Solvers with BDDs
"... Abstract. When BDDs are used for propagation in a constraint solver with nogood recording, it is necessary to find a small subset of a given set of variable assignments that is enough for a BDD to imply a new variable assignment. We show that the task of finding such a minimum subset is NPcomplete ..."
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Cited by 5 (0 self)
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Abstract. When BDDs are used for propagation in a constraint solver with nogood recording, it is necessary to find a small subset of a given set of variable assignments that is enough for a BDD to imply a new variable assignment. We show that the task of finding such a minimum subset is NPcomplete by reduction from the hitting set problem. We present a new algorithm for finding such a minimal subset, which runs in time linear in the size of the BDD representation. In our experiments, the new method is up to ten times faster than the previous method, thereby reducing the solution time by even more than 80%. Due to linear time complexity the new method is able to scale well. 1
On Automata, MDDs and BDDs in Constraint Satisfaction
"... Abstract. In this paper we analyze the relationships between the variants of deterministic finitestate automata (DFAs), multivalued decision diagrams (MDDs) and binary decision diagrams (BDDs) as currently used for compiling constraint satisfaction problems (CSPs). We highlight the limitations and ..."
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Cited by 5 (2 self)
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Abstract. In this paper we analyze the relationships between the variants of deterministic finitestate automata (DFAs), multivalued decision diagrams (MDDs) and binary decision diagrams (BDDs) as currently used for compiling constraint satisfaction problems (CSPs). We highlight the limitations and benefits of using Boolean encodings and BDDs, in comparison to their multivalued counterparts: MDDs and DFAs. In particular, we show the close relationship between these structures when the Boolean encoding of a CSP is using the clustered variable ordering. We also note that, differences between the variants of DFAs and MDDs used in the CSP literature are minor, and appear only due to the removal of redundant nodes in MDDs. We experimentally compare these structures over a set of realworld and random instances. 1
Two Encodings of DNNF Theories
"... Abstract. The paper presents two new compilation schemes of Decomposable Negation Normal Form (DNNF) theories into Conjunctive Normal Form (CNF) and Linear Integer Programming (MIP), respectively. We prove that the encodings have useful properties such as unit propagation on the CNF formula achieves ..."
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Cited by 3 (0 self)
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Abstract. The paper presents two new compilation schemes of Decomposable Negation Normal Form (DNNF) theories into Conjunctive Normal Form (CNF) and Linear Integer Programming (MIP), respectively. We prove that the encodings have useful properties such as unit propagation on the CNF formula achieves domain consistency on the DNNF theory. The approach is evaluated empirically on random as well as realworld CSPproblems. 1
Generating Specialpurpose Stateless Propagators for Arbitrary Constraints
"... Abstract. Given an arbitrary constraint c on n variables with domain size d, we show how to generate a custom propagator that establishes GAC in time O(nd) by precomputing the propagation that would be performed on every reachable subdomain of scope(c). Our propagators are stateless: they store no s ..."
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Cited by 2 (1 self)
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Abstract. Given an arbitrary constraint c on n variables with domain size d, we show how to generate a custom propagator that establishes GAC in time O(nd) by precomputing the propagation that would be performed on every reachable subdomain of scope(c). Our propagators are stateless: they store no state between calls, and so incur no overhead in storing and backtracking state during search. The preprocessing step can take exponential time and the custom propagator is potentially exponential in size. However, for small constraints used repeatedly, in one problem or many, this technique can provide substantial practical gains. Our experimental results show that, compared with optimised implementations of the table constraint, this technique can lead to an order of magnitude speedup, while doing identical search on realistic problems. The technique can also be many times faster than a decomposition into primitive constraints in the Minion solver. Propagation is so fast that, for constraints available in our solver, the generated propagator compares well with a humanoptimised propagator for the same constraint. 1
Generic Global Constraints based on MDDs
, 2007
"... Abstract. Constraint Programming (CP)[1] has been successfully applied to both constraint satisfaction and constraint optimization problems. A wide variety of specialized global constraints provide critical assistance in achieving a good model that can take advantage of the structure of the problem ..."
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Abstract. Constraint Programming (CP)[1] has been successfully applied to both constraint satisfaction and constraint optimization problems. A wide variety of specialized global constraints provide critical assistance in achieving a good model that can take advantage of the structure of the problem in the search for a solution. However, a key outstanding issue is the representation of ’adhoc ’ constraints that do not have an inherent combinatorial nature, and hence are not modelled well using narrowly specialized global constraints. We attempt to address this issue by considering a hybrid of search and compilation. Specifically we suggest the use of Reduced Ordered MultiValued Decision Diagrams (ROMDDs) as the supporting data structure for a generic global constraint. We give an algorithm for maintaining generalized arc consistency (GAC) on this constraint that amortizes the cost of the GAC computation over a roottoleaf path in the search tree without requiring asymptotically more space than used for the MDD. Furthermore we present an approach for incrementally maintaining the reduced property of the MDD during the search, and show how this can be used for providing domain entailment detection. Finally we discuss how to apply our approach to other similar data structures such as AOMDDs and Case DAGs. The technique used can be seen as an extension of the GAC algorithm for the regular language constraint on finite length input [2]. 1
Implementation and Applications of NonBinary Ad Hoc Constraints
, 2008
"... Many reallife combinatorial problems such as scheduling and planning can be modeled as a constraint satisfaction problem (CSP). Informally speaking, a CSP comprises a set of constraints over a set of variables, where each variable can only take values from its domain which is, without loss of gener ..."
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Many reallife combinatorial problems such as scheduling and planning can be modeled as a constraint satisfaction problem (CSP). Informally speaking, a CSP comprises a set of constraints over a set of variables, where each variable can only take values from its domain which is, without loss of generality, a set of nonnegative integers. Solving a CSP requires finding a value for each variable from its domain so that all constraints are satisfied. Example 1. For the nqueens problem, we need to place n queens on an n × n chess board so that no two queens attack each other. Recall that a queen can move along any row, any column and any diagonal. To model the problem as a CSP, we create n variables x1,...,xn. Each xi denotes the column position of the queen on row i. The domain of xi is therefore {1,...,n}. This implicitly guarantees that no two queens are on the same column, because a variable can take exactly one value from its domain. The constraints which enforce any two queens to be on different columns and diagonals are respectively xi ̸ = xj xj − xi  ̸ = j − i for each 1 ≤ i < j ≤ n. Figure 1 shows a solution of the 4queens problem. To solve a CSP, we can incrementally construct a search tree that explores the (complete) search space. Backtracking occurs when the current branch leads to no solution. Due to NPcompleteness, solving a CSP with backtracking search takes exponential time in the worst case. To reduce the search space, some level of (local) consistency is maintained during search [15]; namely, when a variable has been assigned, a consistency algorithm is executed to reduce the domains of other unassigned variables, 1 x1 Q x2