Combarro, Elías F. and Miranda Menéndez, Pedro
(2008)
*On the polytope of non-additive measures.*
Fuzzy Sets and Systems, 159
(16).
pp. 2145-2162.
ISSN 0165-0114

PDF
Restringido a Repository staff only hasta 2020. 822kB |

Official URL: http://www.sciencedirect.com/science/article/pii/S016501140700557X

## Abstract

In this paper we deal with the problem of studying the structure of the polytope of non-additive measures for finite referential sets. We give a necessary and sufficient condition for two extreme points of this polytope to be adjacent. We also show that it is possible to find out in polynomial time whether two vertices are adjacent. These results can be extended to the polytope given by the convex hull of monotone Boolean functions. We also give some results about the facets and edges of the polytope of non-additive measures; we prove that the diameter of the polytope is 3 for referentials of three elements or more. Finally, we show that the polytope is combinatorial and study the corresponding properties; more concretely, we show that the graph of non-additive measures is Hamilton connected if the cardinality of the referential set is not 2.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Non-additive measures; Monotone Boolean functions; Adjacency; Complexity; Diameter; Combinatorial polytopes; Stack filters |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 17033 |

References: | M. Aigner, Combinatorial Theory, Springer, Berlin, 1979. M. Allais, Le comportement de l’homme rationnel devant le risque: critique des postulats de l’école américaine, Econometrica 21 (1953) 503–546 (in French). F.J. Anscombe, R.J. Aumann, A definition of subjective probability, Ann. Math. Statist. 34 (1963) 199–205. A. Chateauneuf, Modelling attitudes towards uncertainty and risk through the use of Choquet integral, Ann. Oper. Res.52 (1994) 3–20. G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953) 131–295. E.F. Combarro, P. Miranda, Identification of fuzzy measures from sample data with genetic algorithms, Comput. Oper.Res. 33 (10) (2006) 3046–3066. R. Dedekind, Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler, Festschrift Hoch Braunschweig Ges. Werke II (1897)103–148 (in German). A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist. 38 (1967) 325–339. D. Denneberg, Non-additive Measures and Integral, Kluwer Academic Publishers, Dordrecht, 1994. D. Ellsberg, Risk, ambiguity, and the savage axioms, Quart. J. Econom. 75 (1961) 643–669. K. Engel, Sperner Theory, Cambridge University Press, Cambridge, 1997. R. Fidytek, A.W. Mostowski, R. Somla, A. Szepietowski, Algorithms counting monotone Boolean functions, Inform. Process. Lett. 79 (5) (2001) 203–209. M. Garey, D. Johnson, Computers and Intractability: A Guide to the Theory NP-Completeness, Mc-Graw Hill, New York, 1979. D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA, 1989. M. Grabisch, Alternative representations of discrete fuzzy measures for decision making, Internat. J. Uncertainty, Fuzziness Knowledge Based Systems 5 (1997) 587–607. M. Grabisch, k-order additive discrete fuzzy measures and their representation, Fuzzy Sets and Systems 92 (1997) 167–189. P.L. Hammer, R. Holzman, On approximations of pseudo-Boolean functions, Zeitschrift für Oper. Res. Math. Methods Oper. Res. 36 (1992) 3–21. A.D. Korshunov, Monotone Boolean functions, Russian Math. Surveys 58 (5) (2003) 929–1001. T. Matsui, NP-completeness of non-adjacency relations on some 0–1-polytopes, In: Lecture Notes in Operations Research, Vol. 1, 1995, pp. 249–258. P. Miranda, E.F. Combarro, On the structure of some families of fuzzy measures, IEEE Trans. Fuzzy Systems 15 (6) (2007) 1068–1081. P. Miranda, E.F. Combarro, P. Gil, Extreme points of some families of non-additive measures, European J. Oper. Res. 33 (10) (2006) 3046–3066. D. Naddef, W.R. Pulleyblank, Hamiltonicity and combinatorial polyhedra, J. Combin. Theory Ser. B 31 (1981) 297–312. C. Papadimitriou, The adjacency relation on the travelling salesman polytope is NP-complete, Math. Programming 14 (1) (1978). D. Radojevic, The logical representation of the discrete Choquet integral, Belg. J. Oper. Res. Statist. Comput. Sci. 38 (2–3) (1998) 67–89. K. Rosen, Discrete Mathematics and Its Applications, McGraw Hill, New York, 1988. G.C. Rota, On the foundations of combinatorial theory I. Theory of Möbius functions, Zeitschrift fürWahrscheinlichkeitstheorie und Verwandte Gebiete 2 (1964) 340–368. D. Schmeidler, Integral representation without additivity, Proc. Amer. Math. Soc. 97 (2) (1986) 255–261. L.S. Shapley, A value for n-person games, in: H.W. Kuhn, A.W. Tucker (Eds.), Contributions to the Theory of Games, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1953, Vol. II, pp. 307–317. I. Shmulevich, T.M. Selke, E.J. Coyle, Stack filters and free distributive lattices, in: Proc. 1995 IEEEWorkshop on Nonlinear Signal Processing,Halkidiki, Greece, June 1995, pp. 927–930. L.A. Skornjakov, Elements of Lattice Theory, Adam Hilger, Bristol, 1977. M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Thesis, Tokyo Institute of Technology, 1974. J. von Neumann, O. Morgenstern, Theory of Games and Economic Behaviour, Princeton University Press, NJ, USA, 1944. P. Wendt, E. Coyle, N.J. Gallagher, Stack filters, IEEE Trans. Acoust. Speech Signal Process. 1986, pp. 898–911 |

Deposited On: | 06 Nov 2012 12:22 |

Last Modified: | 07 Feb 2014 09:40 |

Repository Staff Only: item control page