Generating highly balanced sudoku problems as hard problems
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Sudoku problems are some of the most known and enjoyed pastimes, with a never diminishing popularity, but, for the last few years those problems have gone from an entertainment to an interesting research area, a twofold interesting area, in fact. On the one side Sudoku problems, being a variant of Gerechte Designs and Latin Squares, are being actively used for experimental design, as in [8, 44, 39, 9]. On the other hand, Sudoku problems, as simple as they seem, are really hard structured combinatorial search problems, and thanks to their characteristics and behavior, they can be used as benchmark problems for refining and testing solving algorithms and approaches. Also, thanks to their high inner structure, their study can contribute more than studies of random problems to our goal of solving real-world problems and applications and understanding problem characteristics that make them hard to solve. In this work we use two techniques for solving and modeling Sudoku problems, namely, Constraint Satisfaction Problem (CSP) and Satisfiability Problem (SAT) approaches. To this effect we define the Generalized Sudoku Problem (GSP), where regions can be of rectangular shape, problems can be of any order, and solution existence is not guaranteed. With respect to the worst-case complexity, we prove that GSP with block regions of m rows and n columns with m = n is NP-complete. For studying the empirical hardness of GSP, we define a series of instance generators, that differ in the balancing level they guarantee between the constraints of the problem, by finely controlling how the holes are distributed in the cells of the GSP. Experimentally, we show that the more balanced are the constraints, the higher the complexity of solving the GSP instances, and that GSP is harder than the Quasigroup Completion Problem (QCP), a problem generalized by GSP. Finally, we provide a study of the correlation between backbone variables – variables with the same value in all the solutions of an instance– and hardness of GSP.
Is part ofJournal of Heuristics, 2011, vol. 17, núm. 5, p.589–614
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Ansótegui Gil, Carlos José; Béjar Torres, Ramón; Fernàndez Camon, César; Gomes, Carla; Mateu Piñol, Carles (Association for the Advancement of Artificial Intelligence, 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 ourunderstanding of problem hardness, beyond standard worst-case ...
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Béjar Torres, Ramón; Fernàndez Camon, César; Mateu Piñol, Carles; Valls Marsal, Magda (Elsevier, 2012)The sudoku completion problem is a special case of the latin square completion problem and both problems are known to be NP-complete. However, in the case of a rectangular hole pattern – i.e. each column (or row) is ...