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Kakuro as a Constraint Problem
"... In this paper we describe models of the logic puzzle Kakuro as a constraint problem with finite domain variables. We show a basic model expressing the constraints of the problem and present various improvements to enhance the constraint propagation, and compare alternatives using MILP and SAT solve ..."
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Cited by 45 (1 self)
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In this paper we describe models of the logic puzzle Kakuro as a constraint problem with finite domain variables. We show a basic model expressing the constraints of the problem and present various improvements to enhance the constraint propagation, and compare alternatives using MILP and SAT solvers. Results for different puzzle collections are given. We also propose a grading scheme predicting the difficulty of a puzzle for a human and show how problems can be tightened by removing hints.
Toolsupport for the analysis of hybrid systems and models
 in Proc. of DATE
, 2007
"... This paper introduces a method and toolsupport for the automatic analysis and verification of hybrid and embedded control systems, whose continuous dynamics are often modelled using MATLAB/Simulink. The method is based upon converting system models into the uniform input language of our efficient m ..."
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Cited by 19 (1 self)
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This paper introduces a method and toolsupport for the automatic analysis and verification of hybrid and embedded control systems, whose continuous dynamics are often modelled using MATLAB/Simulink. The method is based upon converting system models into the uniform input language of our efficient multidomain constraint solving library, ABSOLVER, which is then used for subsequent analysis. Basically, ABSOLVER is an extensible SMTsolver which addresses mixed Boolean and (nonlinear) arithmetic constraint problems as they appear in the design of hybrid control systems. It allows the integration and semantic connection of various domain specific solvers via a logical circuit, such that almost arbitrary multidomain constraint problems can be formulated and solved. Its design has been tailored for extensibility, and thus facilitates the reuse of expert knowledge, in that the most appropriate solver for a given task can be integrated and used. As such the only constraint over the problem domain is the capability of the employed solvers. Our approach to systems verification has been validated in an industrial case study using the model of a car’s steering control system. However, additional benchmarks show that other hard instances of problems could also be solved by ABSOLVER in respectable time, and that for some instances, ABSOLVER’s approach was the only means of solving a problem at all. 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.
Measuring the Hardness of SAT Instances
, 2008
"... The search of a precise measure of what hardness of SAT instances means for stateoftheart solvers is a relevant research question. Among others, the space complexity of treelike resolution (also called hardness), the minimal size of strong backdoors and of cyclecutsets, and the treewidth can be ..."
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The search of a precise measure of what hardness of SAT instances means for stateoftheart solvers is a relevant research question. Among others, the space complexity of treelike resolution (also called hardness), the minimal size of strong backdoors and of cyclecutsets, and the treewidth can be used for this purpose. We propose the use of the treelike space complexity as a solid candidate to be the best measure for solvers based on DPLL. To support this thesis we provide a comparison with the other mentioned measures. We also conduct an experimental investigation to show how the proposed measure characterizes the hardness of random and industrial instances.
The graph grammar library  a generic framework for chemical graph rewrite systems
 THEORY AND PRACTICE OF MODEL TRANSFORMATIONS, PROC. OF ICMT 2013, VOLUME 7909 OF LNCS
, 2013
"... Graph rewrite systems are powerful tools to model and study complex problems in various fields of research. Their successful application to chemical reaction modelling on a molecular level was shown but no appropriate and simple system is available at the moment. The presented Graph Grammar Library ..."
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Cited by 5 (2 self)
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Graph rewrite systems are powerful tools to model and study complex problems in various fields of research. Their successful application to chemical reaction modelling on a molecular level was shown but no appropriate and simple system is available at the moment. The presented Graph Grammar Library (GGL) implements a generic Double Push Out approach for general graph rewrite systems. The framework focuses on a high level of modularity as well as high performance, using stateoftheart algorithms and data structures, and comes with extensive documentation. The large GGL chemistry module enables extensive and detailed studies of chemical systems. It well meets the requirements and abilities envisioned by Yadav et al. (2004) for such chemical rewrite systems. Here, molecules are represented as undirected labeled graphs while chemical reactions are described by according graph grammar rules. Beside the graph transformation, the GGL offers advanced cheminformatics algorithms for instance to estimate energies ofmolecules or aromaticity perception. These features are illustrated using a set of reactions from polyketide chemistry a huge class of natural compounds of medical relevance. The graph grammar based simulation of chemical reactions offered by the GGL is a powerful tool for extensive cheminformatics studies on a molecular level. The GGL already provides rewrite rules for all enzymes listed in the KEGG LIGAND database is freely available at
Solving Sudoku Using Combined Message Passing Algorithms
"... In this paper we apply messagepassing algorithms to solve Sudoku puzzles. We provide explicit expression for the sumproduct algorithm and the maxproduct algorithm and analyze the difference between the algorithms in terms of performance and efficiency. The failure of the maxproduct algorithm whe ..."
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Cited by 4 (0 self)
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In this paper we apply messagepassing algorithms to solve Sudoku puzzles. We provide explicit expression for the sumproduct algorithm and the maxproduct algorithm and analyze the difference between the algorithms in terms of performance and efficiency. The failure of the maxproduct algorithm when been applied to Sudoku problem is due to the existence of stoppingsets. We show empirically that the performance of the maxproduct algorithm in the case of Sudoku can be improved by adding redundant constraints. We show that applying the sumproduct to the stoppingset obtained by the maxproduct, can lead to effective and efficient puzzlesolving method.
The Solution of SAT Problems Using Ternary Vectors and Parallel Processing
"... Abstract—This paper will show a new approach to the solution of SATproblems. It has been based on the isomorphism between the Boolean algebras of finite sets and the Boolean algebras of logic functions depending on a finite number of binary variables. Ternary vectors are the main data structure rep ..."
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Abstract—This paper will show a new approach to the solution of SATproblems. It has been based on the isomorphism between the Boolean algebras of finite sets and the Boolean algebras of logic functions depending on a finite number of binary variables. Ternary vectors are the main data structure representing sets of Boolean vectors. The respective set operations (mainly the complement and the intersection) can be executed in a bitparallel way (64 bits at present), but additionally also on different processors working in parallel. Even a hierarchy of processors, a small set of processor cores of a single CPU, and the huge number of cores of the GPU has been taken into consideration. There is no need for any search algorithms. The approach always finds all solutions of the problem without consideration of special cases (such us no solution, one solution, all solutions). It also allows to include problemrelevant knowledge into the problemsolving process at an early point of time. Very often it is possible to use ternary vectors directly for the modeling of a problem. Some examples are used to illustrate the efficiency of this approach (Sudoku, Queen’s problems on the chessboard, node bases in graphs, graphcoloring problems, Hamiltonian and Eulerian paths etc.). Keywords—SAT–solver, ternary vector, parallel processing, XBOOLE I.
Heuristic Reasoning on Graph and Game Complexity of Sudoku
, 903
"... The Sudoku puzzle has achieved worldwide popularity recently, and attracted great attention of the computational intelligence community. Sudoku is always considered as Satisfiability Problem or Constraint Satisfaction Problem. In this paper, we propose to focus on the essential graph structure under ..."
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The Sudoku puzzle has achieved worldwide popularity recently, and attracted great attention of the computational intelligence community. Sudoku is always considered as Satisfiability Problem or Constraint Satisfaction Problem. In this paper, we propose to focus on the essential graph structure underlying the Sudoku puzzle. First, we formalize Sudoku as a graph. Then a solving algorithm based on heuristic reasoning on the graph is proposed. The related rReduction theorem, inference theorem and their properties are proved, providing the formal basis for developments of Sudoku solving systems. In order to evaluate the difficulty levels of puzzles, a quantitative measurement of the complexity level of Sudoku puzzles based on the graph structure and information theory is proposed. Experimental results show that all the puzzles can be solved fast using the proposed heuristic reasoning, and that the proposed game complexity metrics can discriminate difficulty levels of puzzles perfectly. 1.
SUDOKUSAT—A Tool for Analyzing Difficult Sudoku Puzzles
"... Sudoku puzzles enjoy worldwide popularity, and a large community of puzzlers is hoping for ever more difficult puzzles. A crucial step for generating difficult Sudoku puzzles is the fast assessment of the difficulty of a puzzle. In a study in 2006, it has been shown that SAT solving provides a way ..."
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Sudoku puzzles enjoy worldwide popularity, and a large community of puzzlers is hoping for ever more difficult puzzles. A crucial step for generating difficult Sudoku puzzles is the fast assessment of the difficulty of a puzzle. In a study in 2006, it has been shown that SAT solving provides a way to efficiently differentiate between Sudoku puzzles according to their difficulty, by analyzing which resolution technique solves a given puzzle. This paper shows that one of these techniques—unit resolution with failed literal propagation—does not solve a recently published Sudoku puzzle called AI Escargot, claimed to be the world’s most difficult. The technique is also unable to solve any of a list of difficult puzzles published after AI Escargot, whereas it solves all previously studied Sudoku puzzles. We show that the technique can serve as an efficient and reliable computational method for distinguishing the most difficult Sudoku puzzles. As a proofofconcept for an efficient difficulty checker, we present the tool SUDOKUSAT that categorizes Sudoku puzzles with respect to the resolution technique required for solving them. 1. Sudoku Sudoku puzzles have fascinated puzzle solvers since their invention in 1979. A Sudoku puzzle is a 9 × 9 grid of cells, which is composed of nine 3 × 3 nonoverlapping boxes of cells. The objective is to fill the grid with digits from 1 to 9 so that each row, column and box contains any digit at most once. Some cells are already filled with digits; these cells are called hints. To be a proper Sudoku puzzle, the grid must admit a unique way to fill the remaining cells to meet the objective (Uniqueness Property). Figure 1 shows a Sudoku puzzle that can be solved within a few minutes by an experienced puzzler. Whereas precursors of Sudoku—based on magic squares and Latin squares—were known since the late 19 th century,