International Journal of Theoretical and Applied Mathematics
Volume 3, Issue 2, April 2017, Pages: 82-87
Received: Oct. 28, 2016;
Accepted: Jan. 12, 2017;
Published: Mar. 2, 2017
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Xavier Molinero, Department of Mathematics, Universitat Politècnica de Catalunya, Manresa, Spain
We connect three different topics: combinatorial structures, game theory and chemistry. In particular, we establish the bases to represent some simple games, defined as influence games, and molecules, defined from atoms, by using combinatorial structures. First, we characterize simple games as influence games using influence graphs. It let us to modeling simple games as combinatorial structures (from the viewpoint of structures or graphs). Second, we formally define molecules as combinations of atoms. It let us to modeling molecules as combinatorial structures (from the viewpoint of combinations). It is open to generate such combinatorial structures using some specific techniques as genetic algorithms, (meta-) heuristics algorithms and parallel programming, among others.
Combinatorial Structures to Construct Simple Games and Molecules, International Journal of Theoretical and Applied Mathematics.
Vol. 3, No. 2,
2017, pp. 82-87.
M. Akhmedov, I. Kwee, and R. Montemanni. A divide and conquer matheuristic algorithm for the prizecollecting steiner tree problem. Computers & Operations Research, 70: 18–25, 2016.
E. Altman, T. Boulogne, R. El-Azouzi, T. Jimnez, and L. Wynter. A survey on networking games in telecommunications. Computers & Operations Research, 33 (2): 286–311, 2006. Game Theory: Numerical Methods and ApplicationsGame Theory: Numerical Methods and Applications.
B. Bollobás. Modern graph theory, volume 184 of Graduate Texts in Mathematics. Springer-Verlag, New York, NY, 1998.
T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to Algorithms. The MIT Press, The Massachusetts Institute of Technology, 1990.
P. Flajolet. Mathematical methods in the analysis of algorithms and data structures. Trends in Theoretical Computer Science, pages 225–304, 1988.
P. Flajolet and B. Salvy. Computer algebra libraries for combinatorial structures. J. Symbolic Computation, 20: 653–671, 1995.
P. Flajolet, B. Salvy, and P. Zimmermann. Lambdaupsilon-omega: The 1989 cookbook. Technical Report 1073, INRIA, 1989.
P. Flajolet and R. Sedgewick. The average case analysis of algorithms: Counting and generating functions. Technical Report 1888, INRIA, 1993.
P. Flajolet and J. S. Vitter. Average-case Analysis of Algorithms and Data Structures. In J. Van Leeuwen, editor, Handbook of Theoretical Computer Science, chapter 9. North-Holland, 1990.
P. Flajolet, P. Zimmermann, and B. Van Cutsem. A calculus for the random generation of combinatorial structures. Theoretical Computer Science, 132 (1): 1–35, 1994.
L. A. Goldberg and D. M. Jackson. Combinatorial Enumeration. John Wiley & Sons, 1983.
M. Granovetter. Threshold models of collective behavior. American Journal of Sociology, 83 (6): 1420–1443, 1978.
M. A. El-Sharkawi and K. Y. Lee. Modern Heuristic Optimization Techniques: Theory and Applications to Power Systems. Wiley-IEEE Press.
C. Martínez and X. Molinero. A generic approach for the unranking of labeled combinatorial classes. Random Structures & Algorithms, 19 (3-4): 472–497, 2001.
C. Martínez and X. Molinero. Efficient iteration in admissible combinatorial classes. Theoretical Computer Science, 346 (2–3): 388–417, November 2005.
M. Mitchell. An Introduction to Genetic Algorithms (Complex Adaptive Systems). The MIT Press.
X. Molinero, M. Olsen, and M. Serna. On the complexity of exchanging. Information Processing Letters, 116 (6): 437–441, 2016.
X. Molinero, F. Riquelme, and M. J. Serna. Cooperation through social influence. European Journal of Operation Research, 242 (3): 960–974, May 2015.
X. Molinero and J. Vives. Unranking algorithms for combinatorial structures. International Journal of Applied Mathematics and Informatics, 9: 110–115, 2015.
X. Molinero and J. Vives. Unranking algorithms for combinatorial structures. In M. V. Shitikova N. E. Mastorakis, I. Rudas and Y. S. Shmaliy, editors, Proceedings of the International Conference on Applied Mathematics and Computational Methods in Engineering (AMCME 2015), pages 98–101, 2015.
A. K. Nandi and H. R. Medal. Methods for removing links in a network to minimize the spread of infections. Computers & Operations Research, 69: 10–24, 2016.
T. Schelling. Micromotives and macrobehavior. Fels lectures on public policy analysis. W. W. Norton & Company, New York, NY, 1978.
R. Sedgewick and P. Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, Reading, MA, 1996.
E. Talbi. Metaheuristics: From Design to Implementation. Wiley.
A. Taylor and W. Zwicker. Simple games: Desirability relations, trading, pseudoweightings. Princeton University Press, Princeton, NJ, 1999.
A. D. Taylor and W. S. Zwicker. Simple games: desirability relations, trading, and pseudoweightings. Princeton University Press, New Jersey, USA, 1999.
P. Vasant. Meta-Heuristics Optimization Algorithms in Engineering, Business, Economics, and Finance. IGI Global.
R. Keller, W. Banzhaf, P. Nordin, and F. Francone. Genetic Programming An Introduction. San Francisco, CA: Morgan Kaufmann, 1998.
S. Yazdanfar and M. Sina. Providing a real-time scheduling algorithm for multi-processor systems using the modified colonial competition algorithm. Journal of Current Research in Science, 3 (5): 8–17, 2015.