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Reducing the generalised Sudoku problem to the Hamiltonian cycle problem. (English) Zbl 1354.05018
Summary: The generalised Sudoku problem with \(N\) symbols is known to be NP-complete, and hence is equivalent to any other NP-complete problem, even for the standard restricted version where \(N\) is a perfect square. In particular, generalised Sudoku is equivalent to the, classical, Hamiltonian cycle problem. A constructive algorithm is given that reduces generalised Sudoku to the Hamiltonian cycle problem, where the resultant instance of Hamiltonian cycle problem is sparse, and has \(O(N^3)\) vertices. The Hamiltonian cycle problem instance so constructed is a directed graph, and so a (known) conversion to undirected Hamiltonian cycle problem is also provided so that it can be submitted to the best heuristics. A simple algorithm for obtaining the valid Sudoku solution from the Hamiltonian cycle is provided. Techniques to reduce the size of the resultant graph are also discussed.
MSC:
05B15 Orthogonal arrays, Latin squares, Room squares
05C45 Eulerian and Hamiltonian graphs
05C38 Paths and cycles
Software:
Concorde; LKH
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References:
[1] Yato, T.; Seta, T., Complexity and completeness of finding another solution and its application to puzzles, IEICE Trans. Fundam. Electron., E86-A, 5, 1052-1060, (2003)
[2] H. Simonis, Sudoku as a constraint problem, In: CP Workshop of Modeling and Reformulating Constraint Satisfaction Problems, 2005, pp. 13-27.
[3] I. Lynce, J. Ouaknine, Sudoku as a SAT Problem. In: Proceedings of the 9th Symposium on Artificial Intelligence and Mathematics, 2006.
[4] Bartlett, A.; Chartier, T. P.; Langville, A. V.; Rankin, T. D., An integer programming model for the sudoku problem, J. Online Math. Appl., 8, (2008), Article ID 1798
[5] Karp, R. M., Reducibility among combinatorial problems, (1972), Springer New York · Zbl 0366.68041
[6] Dewdney, A. K., Linear transformations between combinatorial problems, Int. J. Comput. Math., 11, 91-110, (1982) · Zbl 0478.68071
[7] Creignou, N., The class of problems that are linearly equivalent to satisfiability or a uniform method for proving np-completeness, Lect. Notes. Comput. Sci., 145, 111-145, (1995) · Zbl 0874.68127
[8] Filar, J. A.; Haythorpe, M.; Taylor, R., Linearly-growing reductions of karp’s 21 NP-complete problems, Numer. Algebra Control Optim., (2016), in press
[9] Filar, J. A.; Haythorpe, M., A linearly-growing conversion from the set splitting problem to the directed Hamiltonian cycle problem, (Optimization and Control Methods in Industrial Engineering and Construction, (2014)), 35-52
[10] Ejov, V.; Haythorpe, M.; Rossomakhine, S., A linear-size conversion of HCP to 3HCP, Australas. J. Combin., 62, 1, 45-58, (2015) · Zbl 1321.05144
[11] Chalaturnyk, A., A fast algorithm for finding Hamilton cycles, (2008), University of Manitoba, (Masters thesis)
[12] Baniasadi, P.; Ejov, V.; Filar, J. A.; Haythorpe, M.; Rossomakhine, S., Deterministic “snakes and ladders” heuristic for the Hamiltonian cycle problem, Math. Program. Comput., 6, 1, 55-75, (2014) · Zbl 1301.05326
[13] D.L. Applegate, R.B. Bixby, V. Chavátal, W.J. Cook, Concorde TSP Solver, 2015. http://www.tsp.gatech.edu/concorde/index.html (Accessed Jan 20, 2016).
[14] Helsgaun, K., An effective implementation of lin-kernighan traveling salesman heuristic, European J. Oper. Res., 126, 106-130, (2000) · Zbl 0969.90073
[15] A. Johnson, Quasi-linear reduction of Hamiltonian Cycle Problem (HCP) to Satisfiability Problem (SAT). IP.com Disclosure Number IPCOM000237123D, 2014.
[16] Haythorpe, M., Constructing arbitrarily large graphs with a specified number of Hamiltonian cycles, Electron. J. Graph Theory Appl., 4, 1, 18-25, (2016)
[17] M. Haythorpe, FHCP Challenge Set., 2015. http://fhcp.edu.au/fhcpcs.
[18] Newcombe, A., Investigating Hamilton cycles using logistic regression, (2015), (Honours thesis)
[19] Filar, J. A.; Haythorpe, M.; Rossomakhine, S., A new heuristic for detecting non-Hamiltonicity in cubic graphs, Comput. Oper. Res., 64, 283-292, (2015) · Zbl 1349.90813
[20] G. Royle, Minimum Sudoku, 2005. http://staffhome.ecm.uwa.edu.au/ 00013890/sudokumin.php (Accessed 20 January 2016).
[21] McGuire, G.; Tugemann, B.; Civario, G., There is no 16-clue sudoku: solving the sudoku minimum number of clues problem via hitting set enumeration, Exp. Math., 23, 2, 190-217, (2014) · Zbl 1296.05008
[22] B. Knockel, Sudoku Generator, 2013. http://www.mathworks.com/matlabcentral/fileexchange/28168-sudoku-generator (Accessed Sep 16, 2016).
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