A hybrid BDD and SAT finite domain constraint solver (2006)

by P Hawkins, P J Stuckey
Venue:In Proceedings of the 8th International Symposium on Practical Aspects of Declarative Languages, volume 3819 of LNCS
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Propagation via Lazy Clause Generation

by Olga Ohrimenko, Peter J. Stuckey, Michael Codish
"... Finite domain propagation solvers effectively represent the possible values of variables by a set of choices which can be naturally modelled as Boolean variables. In this paper we describe how to mimic a finite domain propagation engine, by mapping propagators into clauses in a SAT solver. This imm ..."
Abstract - Cited by 47 (25 self) - Add to MetaCart
Finite domain propagation solvers effectively represent the possible values of variables by a set of choices which can be naturally modelled as Boolean variables. In this paper we describe how to mimic a finite domain propagation engine, by mapping propagators into clauses in a SAT solver. This immediately results in strong nogoods for finite domain propagation. But a naive static translation is impractical except in limited cases. We show how to convert propagators to lazy clause generators for a SAT solver. The resulting system introduces flexibility in modelling since variables are modelled dually in the propagation engine and the SAT solver, and we explore various approaches to the dual modelling. We show that the resulting system solves many finite domain problems significantly faster than other techniques.
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P.: A Constraint Store Based on Multivalued Decision Diagrams

by H. R. Andersen, T. Hadzic, J. N. Hooker, P. Tiedemann - 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 limited-width multivalued decision diagram (MDD). It reduces to a traditional domain store when the maximum width is on ..."
Abstract - Cited by 29 (10 self) - Add to MetaCart
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 limited-width 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
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Propagation = lazy clause generation

by Olga Ohrimenko, Peter J. Stuckey, Michael Codish - Proceedings of the 13th International Conference on Principles and Practice of Constraint Programming, LNCS , 2007
"... Abstract. Finite domain propagation solvers effectively represent the possible values of variables by a set of choices which can be naturally modelled as Boolean variables. In this paper we describe how we can mimic a finite domain propagation engine, by mapping propagators into clauses in a SAT sol ..."
Abstract - Cited by 23 (12 self) - Add to MetaCart
Abstract. Finite domain propagation solvers effectively represent the possible values of variables by a set of choices which can be naturally modelled as Boolean variables. In this paper we describe how we can mimic a finite domain propagation engine, by mapping propagators into clauses in a SAT solver. This immediately results in strong nogoods for finite domain propagation. But a naive static translation is impractical except in limited cases. We show how we can convert propagators to lazy clause generators for a SAT solver. The resulting system can solve scheduling problems significantly faster than generating the clauses from scratch, or using Satisfiability Modulo Theories solvers with difference logic. 1
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Fast set bounds propagation using a BDD-SAT hybrid

by Graeme Gange, Peter J. Stuckey, Vitaly Lagoon - JOURNAL OF ARTIFICIAL INTELLIGENCE RESEARCH
"... Binary Decision Diagram (BDD) based set bounds propagation is a powerful approach to solving set-constraint satisfaction problems. However, prior BDD based techniques incur the significant overhead of constructing and manipulating graphs during search. We present a set-constraint solver which combin ..."
Abstract - Cited by 9 (5 self) - Add to MetaCart
Binary Decision Diagram (BDD) based set bounds propagation is a powerful approach to solving set-constraint satisfaction problems. However, prior BDD based techniques incur the significant overhead of constructing and manipulating graphs during search. We present a set-constraint solver which combines BDD-based set-bounds propagators with the learning abilities of a modern SAT solver. Together with a number of improvements beyond the basic algorithm, this solver is highly competitive with existing propagation based set constraint solvers.
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Efficient Reasoning for Nogoods in Constraint Solvers with BDDs

by Sathiamoorthy Subbarayan
"... 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 NP-complete ..."
Abstract - Cited by 6 (0 self) - Add to MetaCart
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 NP-complete 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
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Two Encodings of DNNF Theories

by Jean Christoph Jung, Pedro Barahona, George Katsirelos, Toby Walsh
"... 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 ..."
Abstract - Cited by 5 (0 self) - Add to MetaCart
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 real-world CSP-problems. 1
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Combining binary decision diagrams and backtracking search for scalable backtrackfree interactive product configuration

by Andreas Hau Nørgaard, Morten Riiskjær Boysen, Rune Møller Jensen, Peter Tiedemann - in In Proceedings of the IJCAI09 Workshop on Configuration , 2000
"... This paper demonstrates how to create approxima-tions of a configuration problem using BDDs to im-prove performance over a pure search-based con-figurator, for problems that are intractable to rep-resent by a monolithic BDD. The paper demon-strates several ways to build the approximations and it sho ..."
Abstract - Cited by 4 (1 self) - Add to MetaCart
This paper demonstrates how to create approxima-tions of a configuration problem using BDDs to im-prove performance over a pure search-based con-figurator, for problems that are intractable to rep-resent by a monolithic BDD. The paper demon-strates several ways to build the approximations and it shows that using search results for building and improving the approximations leads to a sig-nificant performance gain. 1

Fast Set Bounds Propagation using BDDs

by Graeme Gange, Vitaly Lagoon, Peter J. Stuckey , 2008
"... Set bounds propagation is the most popular approach to solving constraint satisfaction problems (CSPs) involving set variables. The use of reduced ordered Binary Decision Diagrams (BDDs) to represent and solve set CSPs is well understood and brings the advantage that propagators for arbitrary set co ..."
Abstract - Cited by 4 (2 self) - Add to MetaCart
Set bounds propagation is the most popular approach to solving constraint satisfaction problems (CSPs) involving set variables. The use of reduced ordered Binary Decision Diagrams (BDDs) to represent and solve set CSPs is well understood and brings the advantage that propagators for arbitrary set constraints can be built. This can substantially improve solving. The disadvantages of BDDs is that creating and manipulating BDDs can be expensive. In this paper we show how we can perform set bounds propagation using BDDs in a much more efficient manner by generically creating set constraint predicates, and using a marking approach to propagation. The resulting system can be significantly faster than competing approaches to set bounds propagation.
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Modelling for Lazy Clause Generation

by Olga Ohrimenko, Peter J. Stuckey
"... Lazy clause generation is a hybrid SAT and finite domain propagation solver that tries to combine the advantages of both: succinct modelling using finite domains and powerful nogoods and backjumping search using SAT technology. It has been shown that it can solve hard scheduling problems significant ..."
Abstract - Cited by 3 (1 self) - Add to MetaCart
Lazy clause generation is a hybrid SAT and finite domain propagation solver that tries to combine the advantages of both: succinct modelling using finite domains and powerful nogoods and backjumping search using SAT technology. It has been shown that it can solve hard scheduling problems significantly faster than SAT or standard finite domain propagation alone. This new hybrid opens up many choices in modelling problems because of its dual representation of problems as both finite domain and SAT variables. In this paper we investigate some of those choices. Arising out of the modelling choices comes a novel combination of bounds representation and domain propagation which creates a form of propagation of disjunctions. We show this novel modelling approach can outperform more standard approaches on some problems. 1
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Pairing Functions, Boolean Evaluation and Binary Decision Diagrams in Prolog

by Paul Tarau , 808
"... Abstract. A “pairing function ” J associates a unique natural number z to any two natural numbers x,y such that for two “unpairing functions” K and L, the equalities K(J(x,y))=x, L(J(x,y))=y and J(K(z),L(z))=z hold. Using pairing functions on natural number representations of truth tables, we derive ..."
Abstract - Cited by 1 (0 self) - Add to MetaCart
Abstract. A “pairing function ” J associates a unique natural number z to any two natural numbers x,y such that for two “unpairing functions” K and L, the equalities K(J(x,y))=x, L(J(x,y))=y and J(K(z),L(z))=z hold. Using pairing functions on natural number representations of truth tables, we derive an encoding for Binary Decision Diagrams with the unique property that its boolean evaluation faithfully mimics its structural conversion to a a natural number through recursive application of a matching pairing function. We then use this result to derive ranking and unranking functions for BDDs and reduced BDDs. The paper is organized as a self-contained literate Prolog program, available at
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