A comparison of several enumerative algorithms for Sudoku
Sudoku is a puzzle played of an n × n grid where n is the square of a positive integer m. The most common size is n=9. The grid is partitioned into n subgrids of size m × m. The player must place exactly one number from the set N={1, ..., n} in each row and each column of as well as in each subgrid....
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| Published in: | The Journal of the Operational Research Society Vol. 65; no. 10; pp. 1602 - 1610 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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London
Taylor & Francis
01.10.2014
Palgrave Macmillan Palgrave Macmillan UK Taylor & Francis Ltd |
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| ISSN: | 0160-5682, 1476-9360 |
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| Abstract | Sudoku is a puzzle played of an n × n grid
where n is the square of a positive integer m. The most common size is n=9. The grid is partitioned into n subgrids of size m × m. The player must place exactly one number from the set N={1, ..., n} in each row and each column of
as well as in each subgrid. A grid is provided with some numbers already in place, called givens. In this paper, some relationships between Sudoku and several operations research problems are presented. We model the problem by means of two mathematical programming formulations. The first one consists of an integer linear programming model, while the second one is a tighter non-linear integer programming formulation. We then describe several enumerative algorithms to solve the puzzle and compare their relative efficiencies. Two basic backtracking algorithms are first described for the general Sudoku. We then solve both formulations by means of constraint programming. Computational experiments are performed to compare the efficiency and effectiveness of the proposed algorithms. Our implementation of a backtracking algorithm can solve most benchmark instances of size 9 within 0.02 s, while no such instance was solved within that time by any other method. Our implementation is also much faster than an existing alternative algorithm. |
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| AbstractList | Sudoku is a puzzle played of an n × n grid N, where n is the square of a positive integer m. The most common size is n = 9. The grid is partitioned into n subgrids of size m × m. The player must place exactly one number from the set N = {1,...,n} in each row and each column of N, as well as in each subgrid. A grid is provided with some numbers already in place, called givens. In this paper, some relationships between Sudoku and several operations research problems are presented. We model the problem by means of two mathematical programming formulations. The first one consists of an integer linear programming model, while the second one is a tighter non-linear integer programming formulation. We then describe several enumerative algorithms to solve the puzzle and compare their relative efficiencies. Two basic backtracking algorithms are first described for the general Sudoku. We then solve both formulations by means of constraint programming. Computational experiments are performed to compare the efficiency and effectiveness of the proposed algorithms. Our implementation of a backtracking algorithm can solve most benchmark instances of size 9 within 0.02 s, while no such instance was solved within that time by any other method. Our implementation is also much faster than an existing alternative algorithm. Sudoku is a puzzle played of an n × n grid (Formula Omitted: See PDF) where n is the square of a positive integer m. The most common size is n=9. The grid is partitioned into n subgrids of size m × m. The player must place exactly one number from the set N={1, ..., n} in each row and each column of (Formula Omitted: See PDF) as well as in each subgrid. A grid is provided with some numbers already in place, called givens. In this paper, some relationships between Sudoku and several operations research problems are presented. We model the problem by means of two mathematical programming formulations. The first one consists of an integer linear programming model, while the second one is a tighter non-linear integer programming formulation. We then describe several enumerative algorithms to solve the puzzle and compare their relative efficiencies. Two basic backtracking algorithms are first described for the general Sudoku. We then solve both formulations by means of constraint programming. Computational experiments are performed to compare the efficiency and effectiveness of the proposed algorithms. Our implementation of a backtracking algorithm can solve most benchmark instances of size 9 within 0.02 s, while no such instance was solved within that time by any other method. Our implementation is also much faster than an existing alternative algorithm. [PUBLICATION ABSTRACT] Sudoku is a puzzle played of an n × n grid where n is the square of a positive integer m . The most common size is n =9. The grid is partitioned into n subgrids of size m × m . The player must place exactly one number from the set N ={1, …, n } in each row and each column of as well as in each subgrid. A grid is provided with some numbers already in place, called givens . In this paper, some relationships between Sudoku and several operations research problems are presented. We model the problem by means of two mathematical programming formulations. The first one consists of an integer linear programming model, while the second one is a tighter non-linear integer programming formulation. We then describe several enumerative algorithms to solve the puzzle and compare their relative efficiencies. Two basic backtracking algorithms are first described for the general Sudoku. We then solve both formulations by means of constraint programming. Computational experiments are performed to compare the efficiency and effectiveness of the proposed algorithms. Our implementation of a backtracking algorithm can solve most benchmark instances of size 9 within 0.02 s, while no such instance was solved within that time by any other method. Our implementation is also much faster than an existing alternative algorithm. Sudoku is a puzzle played of an n × n grid where n is the square of a positive integer m. The most common size is n=9. The grid is partitioned into n subgrids of size m × m. The player must place exactly one number from the set N={1, ..., n} in each row and each column of as well as in each subgrid. A grid is provided with some numbers already in place, called givens. In this paper, some relationships between Sudoku and several operations research problems are presented. We model the problem by means of two mathematical programming formulations. The first one consists of an integer linear programming model, while the second one is a tighter non-linear integer programming formulation. We then describe several enumerative algorithms to solve the puzzle and compare their relative efficiencies. Two basic backtracking algorithms are first described for the general Sudoku. We then solve both formulations by means of constraint programming. Computational experiments are performed to compare the efficiency and effectiveness of the proposed algorithms. Our implementation of a backtracking algorithm can solve most benchmark instances of size 9 within 0.02 s, while no such instance was solved within that time by any other method. Our implementation is also much faster than an existing alternative algorithm. |
| Author | Coelho, Leandro C Laporte, Gilbert |
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| DOI | 10.1057/jors.2013.114 |
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| References | Naveh, Richter, Altshuler, Gresh, Connors (CR18) 2007; 51 Apt (CR2) 2003 CR17 CR16 Lewis (CR13) 2007; 130 CR12 Higgins, Kozan, Ferreira (CR9) 1996; 300 Lustig, Puget (CR14) 2001; 310 Sellmann, Zervoudakis, Stamatopoulos, Fahle (CR23) 2002; 1150 Rossi, Van Beek, Walsh (CR20) 2006 Takayuki, Takahiro (CR26) 2003; 860 Jensen, Toft (CR10) 1994 Golomb, Baumert (CR8) 1965; 120 CR3 CR6 Applegate, Cook (CR1) 1991; 30 Dànes, Keedwell (CR5) 1974 CR7 Laporte, Pesant (CR11) 2004; 55 CR27 CR25 CR24 CR22 Colbourn, Colbourn, Stinson (CR4) 1984 CR21 Mantere, Koljonen (CR15) 2006 Rodgers, Nicewander (CR19) 1988; 420 CIT0001 CIT0023 CIT0011 Dànes J (CIT0005) 1974 Apt K (CIT0002) 2003 Rossi F (CIT0020) 2006 Colbourn CJ (CIT0004) 1984 Takayuki Y (CIT0026) 2003; 860 CIT0014 CIT0013 Mantere T (CIT0015) 2006 CIT0018 CIT0009 Jensen TR (CIT0010) 1994 CIT0008 CIT0019 |
| References_xml | – volume: 120 start-page: 516 issue: 4 year: 1965 end-page: 524 ident: CR8 article-title: Backtrack programming publication-title: Journal of the ACM doi: 10.1145/321296.321300 – year: 1974 ident: CR5 publication-title: Latin Squares and their Applications – ident: CR22 – volume: 300 start-page: 147 issue: 2 year: 1996 end-page: 161 ident: CR9 article-title: Optimal scheduling of trains on a single line track publication-title: Transportation Research Part B: Methodological doi: 10.1016/0191-2615(95)00022-4 – year: 1994 ident: CR10 publication-title: Graph Coloring Problems doi: 10.1002/9781118032497 – start-page: 86 year: 2006 end-page: 92 ident: CR15 article-title: Solving and rating Sudoku puzzles with genetic algorithms publication-title: New Developments in Artificial Intelligence and the Semantic Web-Proceedings of the 12th Finnish Artificial Conference – volume: 51 start-page: 263 issue: 3.4 year: 2007 end-page: 279 ident: CR18 article-title: Workforce optimization: Identification and assignment of professional workers using constraint programming publication-title: IBM Journal of Research and Development doi: 10.1147/rd.513.0263 – ident: CR16 – volume: 310 start-page: 29 issue: 6 year: 2001 end-page: 53 ident: CR14 article-title: Program does not equal program: Constraint programming and its relationship to mathematical programming publication-title: Interfaces doi: 10.1287/inte.31.6.29.9647 – ident: CR12 – volume: 130 start-page: 387 issue: 4 year: 2007 end-page: 401 ident: CR13 article-title: Metaheuristics can solve Sudoku puzzles publication-title: Journal of Heuristics doi: 10.1007/s10732-007-9012-8 – ident: CR6 – volume: 420 start-page: 59 issue: 1 year: 1988 end-page: 66 ident: CR19 article-title: Thirteen ways to look at the correlation coefficient publication-title: The American Statistician doi: 10.2307/2685263 – volume: 860 start-page: 1052 issue: 5 year: 2003 end-page: 1060 ident: CR26 article-title: Complexity and completeness of finding another solution and its application to puzzles publication-title: IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences – ident: CR25 – ident: CR27 – volume: 55 start-page: 1208 issue: 11 year: 2004 end-page: 1217 ident: CR11 article-title: A general multi-shift scheduling system publication-title: Journal of the Operational Research Society doi: 10.1057/palgrave.jors.2601789 – ident: CR21 – year: 2003 ident: CR2 publication-title: Principles of Constraint Programming doi: 10.1017/CBO9780511615320 – ident: CR3 – ident: CR17 – ident: CR7 – volume: 30 start-page: 149 issue: 2 year: 1991 end-page: 156 ident: CR1 article-title: A computational study of the job-shop scheduling problem publication-title: ORSA Journal on Computing doi: 10.1287/ijoc.3.2.149 – ident: CR24 – start-page: 248 year: 1984 end-page: 253 ident: CR4 article-title: The computational complexity of recognizing critical sets publication-title: Graph Theory Singapore 1983, Volume 1073 of Lecture Notes in Mathematics – year: 2006 ident: CR20 publication-title: Handbook of Constraint Programming – volume: 1150 start-page: 207 issue: 1 year: 2002 end-page: 225 ident: CR23 article-title: Crew assignment via constraint programming: Integrating column generation and heuristic tree search publication-title: Annals of Operations Research doi: 10.1023/A:1021105422248 – ident: CIT0009 doi: 10.1016/0191-2615(95)00022-4 – start-page: 248 volume-title: Graph Theory Singapore 1983, Volume 1073 of Lecture Notes in Mathematics year: 1984 ident: CIT0004 – ident: CIT0013 doi: 10.1007/s10732-007-9012-8 – start-page: 86 volume-title: New Developments in Artificial Intelligence and the Semantic Web-Proceedings of the 12th Finnish Artificial Conference year: 2006 ident: CIT0015 – ident: CIT0018 doi: 10.1147/rd.513.0263 – volume-title: Handbook of Constraint Programming year: 2006 ident: CIT0020 – volume-title: Latin Squares and their Applications year: 1974 ident: CIT0005 – ident: CIT0001 doi: 10.1287/ijoc.3.2.149 – ident: CIT0011 doi: 10.1057/palgrave.jors.2601789 – volume-title: Graph Coloring Problems year: 1994 ident: CIT0010 doi: 10.1002/9781118032497 – volume: 860 start-page: 1052 issue: 5 year: 2003 ident: CIT0026 publication-title: IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences – ident: CIT0008 doi: 10.1145/321296.321300 – volume-title: Principles of Constraint Programming year: 2003 ident: CIT0002 doi: 10.1017/CBO9780511615320 – ident: CIT0023 doi: 10.1023/A:1021105422248 – ident: CIT0014 doi: 10.1287/inte.31.6.29.9647 – ident: CIT0019 doi: 10.2307/2685263 |
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| Snippet | Sudoku is a puzzle played of an n × n grid
where n is the square of a positive integer m. The most common size is n=9. The grid is partitioned into n subgrids... Sudoku is a puzzle played of an n × n grid N, where n is the square of a positive integer m. The most common size is n = 9. The grid is partitioned into n... Sudoku is a puzzle played of an n × n grid where n is the square of a positive integer m . The most common size is n =9. The grid is partitioned into n... Sudoku is a puzzle played of an n × n grid (Formula Omitted: See PDF) where n is the square of a positive integer m. The most common size is n=9. The grid is... |
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| SubjectTerms | Algorithms backtracking Business and Management constraint programming Datasets Game theory General Paper General Papers Heuristics Integer programming Integers linear and non-linear integer programming Linear programming Management Mathematical analysis Mathematical models Mathematical programming Mathematical puzzles Modeling Operations research Operations Research/Decision Theory puzzle Puzzles Studies Sudoku |
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| Title | A comparison of several enumerative algorithms for Sudoku |
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