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Sudoku as a SAT problem
 In Proc. of the Ninth International Symposium on Artificial Intelligence and Mathematics
, 2006
"... Sudoku is a very simple and wellknown puzzle that has achieved international popularity in the recent past. This paper addresses the problem of encoding Sudoku puzzles into conjunctive normal form (CNF), and subsequently solving them using polynomialtime propositional satisfiability (SAT) inferenc ..."
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Cited by 25 (0 self)
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Sudoku is a very simple and wellknown puzzle that has achieved international popularity in the recent past. This paper addresses the problem of encoding Sudoku puzzles into conjunctive normal form (CNF), and subsequently solving them using polynomialtime propositional satisfiability (SAT) inference techniques. We introduce two straightforward SAT encodings for Sudoku: the minimal encoding and the extended encoding. The minimal encoding suffices to characterize Sudoku puzzles, whereas the extended encoding adds redundant clauses to the minimal encoding. Experimental results demonstrate that, for thousands of very hard puzzles, inference techniques struggle to solve these puzzles when using the minimal encoding. However, using the extended encoding, unit propagation is able to solve about half of our set of puzzles. Nonetheless, for some puzzles more sophisticated inference techniques are required. 1
Boolean Equipropagation for Concise and Efficient SAT Encodings of Combinatorial Problems
"... We present an approach to propagationbased SAT encoding of combinatorial problems, Boolean equipropagation, where constraints are modeled as Boolean functions which propagate information about equalities between Boolean literals. This information is then applied to simplify the CNF encoding of the ..."
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Cited by 7 (5 self)
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We present an approach to propagationbased SAT encoding of combinatorial problems, Boolean equipropagation, where constraints are modeled as Boolean functions which propagate information about equalities between Boolean literals. This information is then applied to simplify the CNF encoding of the constraints. A key factor is that considering only a small fragment of a constraint model at one time enables us to apply stronger, and even complete, reasoning to detect equivalent literals in that fragment. Once detected, equivalences apply to simplify the entire constraint model and facilitate further reasoning on other fragments. Equipropagation in combination with partial evaluation and constraint simplification provide the foundation for a powerful approach to SATbased finite domain constraint solving. We introduce a tool called BEE (BenGurion Equipropagation Encoder) based on these ideas and demonstrate for a variety of benchmarks that our approach leads to a considerable reduction in the size of CNF encodings and subsequent speedups in SAT solving times. 1.
The impact of balancing on problem hardness in a highly structured domain
 Proc. of 9th International Conference on Theory and Applications of Satisfiability Testing (SAT ’06
, 2006
"... Random problem distributions have played a key role in the study and design of algorithms for constraint satisfaction and Boolean satisfiability, as well as in our understanding of problem hardness, beyond standard worstcase complexity. We consider random problem distributions from a highly structu ..."
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Cited by 7 (2 self)
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Random problem distributions have played a key role in the study and design of algorithms for constraint satisfaction and Boolean satisfiability, as well as in our understanding of problem hardness, beyond standard worstcase complexity. We consider random problem distributions from a highly structured problem domain that generalizes the Quasigroup Completion problem (QCP) and Quasigroup with Holes (QWH), a widely used domain that captures the structure underlying a range of realworld applications. Our problem domain is also a generalization of the wellknown Sudoku puzzle: we consider Sudoku instances of arbitrary order, with the additional generalization that the block regions can have rectangular shape, in addition to the standard square shape. We evaluate the computational hardness of Generalized Sudoku instances, for different parameter settings. Our experimental hardness results show that we can generate instances that are considerably harder than QCP/QWH instances of the same size. More interestingly, we show the impact of different balancing strategies on problem hardness. We also provide insights into backbone variables in Generalized Sudoku instances and how they correlate to problem hardness.
Efficient SAT techniques for absolute encoding of permutation problems: Application to Hamiltonian cycles
 IN: PROCEEDINGS SARA
, 2009
"... We study novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT). Our focus is on combinatorial problems that can be represented as a permutation of n objects, subject to additional constraints. In the case of the Hamiltonian Cycle Problem (HCP), th ..."
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Cited by 4 (1 self)
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We study novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT). Our focus is on combinatorial problems that can be represented as a permutation of n objects, subject to additional constraints. In the case of the Hamiltonian Cycle Problem (HCP), these constraints are that two adjacent nodes in a permutation should also be neighbors in the graph for which we search for a Hamiltonian cycle. We use the absolute SAT encoding of permutations, where for each of the n objects and each of its possible positions in a permutation, a predicate is defined to indicate whether the object is placed in that position. For implementation of this predicate, we compare the direct and logarithmic encodings that have been used previously, against 16 hierarchical parameterizable encodings of which we explore 416 instantiations. We propose the use of enumerative adjacency constraints—that enumerate the possible neighbors of a node in a permutation— instead of, or in addition to the exclusivity adjacency constraints—that exclude impossible neighbors, and that have been applied previously. We study 11 heuristics for efficiently choosing the first node in the Hamiltonian cycle, as well as 8 heuristics for static CNF variable ordering. We achieve at least 4 orders of magnitude average speedup on HCP benchmarks from the phase transition region, relative to the previously used encodings for solving of HCPs via SAT, such that the speedup is increasing with the size of the graphs.
Proceedings of the Eighth Symposium on Abstraction, Reformulation, and Approximation (SARA2009) Efficient SAT Techniques for Absolute Encoding of Permutation Problems: Application to Hamiltonian Cycles
"... We study novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT). Our focus is on combinatorial problems that can be represented as a permutation of n objects, subject to additional constraints. In the case of the Hamiltonian Cycle Problem (HCP), th ..."
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We study novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT). Our focus is on combinatorial problems that can be represented as a permutation of n objects, subject to additional constraints. In the case of the Hamiltonian Cycle Problem (HCP), these constraints are that two adjacent nodes in a permutation should also be neighbors in the original graph for which we search for a Hamiltonian cycle. We use the absolute SAT encoding of permutations, where for each of the n objects and each of its possible positions in a permutation, a predicate is defined to indicate whether the object is placed in that position. For implementation of this predicate, we compare the direct and logarithmic encodings that have been used previously, against 16 hierarchical parameterizable encodings of which we explore 416 instantiations. We propose the use of enumerative adjacency constraints—that enumerate the possible neighbors of a node in a permutation— instead of, or in addition to the exclusivity adjacency constraints—that exclude impossible neighbors, and that have been applied previously. We study 11 heuristics for efficiently choosing the first node in the Hamiltonian cycle, as well as 8 heuristics for static CNF variable ordering. We achieve at least 4 orders of magnitude average speedup on HCP benchmarks from the phase transition region, relative to the previously used encodings for solving of HCPs via SAT, such that the speedup is increasing with the size of the graphs.
Breaking Local Symmetries in Quasigroup Completion Problems
"... Abstract. Symmetry breaking is wellknown as an important technique for reducing the search space when solving combinatorial problems. Symmetries can be broken either during a preprocessing step or during the search. Local symmetries, in contrast with global symmetries, are applied to a problem inst ..."
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Abstract. Symmetry breaking is wellknown as an important technique for reducing the search space when solving combinatorial problems. Symmetries can be broken either during a preprocessing step or during the search. Local symmetries, in contrast with global symmetries, are applied to a problem instance for which there is a partial assignment. Given such assignment, additional symmetries may hold. We perform an experimental study on breaking local symmetries in quasigroup completion problems, a wellstudied problem among combinatorial problems. We break local symmetries by adding additional clauses to a SAT encoding. We conclude that these additional constraints have a puzzling effect on stateoftheart SAT solvers, mainly due to the heuristics being used by these solvers. 1
Design of Parallel Portfolios for SATBased Solving of Hamiltonian Cycle Problems
"... We study portfolios of parallel strategies for Boolean Satisfiability (SAT) based solving of Hamiltonian Cycle Problems (HCPs). The strategies are based on our techniques for relative SAT encoding of permutations with constraints, and exploit: 1) encodings that eliminate half of the ordering Boolean ..."
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We study portfolios of parallel strategies for Boolean Satisfiability (SAT) based solving of Hamiltonian Cycle Problems (HCPs). The strategies are based on our techniques for relative SAT encoding of permutations with constraints, and exploit: 1) encodings that eliminate half of the ordering Boolean variables and twothirds of the transitivity constraints; 2) 12 triangulation heuristics for minimal enumeration of transitivity; 3) 11 heuristics for selecting the first node in the Hamiltonian cycle; 4) inverse transitivity constraints; and 5) exclusivity successor constraints between neighbors. We achieve up to 3 orders of magnitude speedup on random graphs that have Hamiltonian cycles and are in the phase transition region.
On Balanced CSPs with High Treewidth
, 2007
"... Tractable cases of the binary CSP are mainly divided in two classes: constraint language restrictions and constraint graph restrictions. To better understand and identify the hardest binary CSPs, in this work we propose methods to increase their hardness by increasing the balance of both the constra ..."
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Tractable cases of the binary CSP are mainly divided in two classes: constraint language restrictions and constraint graph restrictions. To better understand and identify the hardest binary CSPs, in this work we propose methods to increase their hardness by increasing the balance of both the constraint language and the constraint graph. The balance of a constraint is increased by maximizing the number of domain elements with the same number of occurrences. The balance of the graph is defined using the classical definition from graph theory. In this sense we present two graph models; a first graph model that increases the balance of a graph maximizing the number of vertices with the same degree, and a second one that additionally increases the girth of the graph, because a high girth implies a high treewidth, an important parameter for binary CSPs hardness. Our results show that our more balanced graph models and constraints result in harder instances when compared to typical random binary CSP instances, by several orders of magnitude. Also we detect, at least for sparse constraint graphs, a higher treewidth for our graph models.
The Impact of Balancing on Problem Hardness in a Highly Structured Domain ∗
"... Random problem distributions have played a key role in the study and design of algorithms for constraint satisfaction and Boolean satisfiability, as well as in our understanding of problem hardness, beyond standard worstcase complexity. We consider random problem distributions from a highly struct ..."
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Random problem distributions have played a key role in the study and design of algorithms for constraint satisfaction and Boolean satisfiability, as well as in our understanding of problem hardness, beyond standard worstcase complexity. We consider random problem distributions from a highly structured problem domain that generalizes the Quasigroup Completion problem (QCP) and Quasigroup with Holes (QWH), a widely used domain that captures the structure underlying a range of realworld applications. Our problem domain is also a generalization of the wellknown Sudoku puzzle: we consider Sudoku instances of arbitrary order, with the additional generalization that the block regions can have rectangular shape, in addition to the standard square shape. We evaluate the computational hardness of Generalized Sudoku instances, for different parameter settings. Our experimental hardness results show that we can generate instances that are considerably harder than QCP/QWH instances of the same size. More interestingly, we show the impact of different balancing strategies on problem hardness. We also provide insights into backbone variables in Generalized Sudoku instances and how they correlate to problem hardness.