Abstract
Two classes of competitive facility location models are considered, in which several persons (players) sequentially or simultaneously open facilities for serving clients. The first class consists of discrete two-level programming models. The second class consists of game models with several independent players pursuing selfish goals. For the first class, its relationship with pseudo-Boolean functions is established and a novel method for constructing a family of upper and lower bounds on the optimum is proposed. For the second class, the tight PLS-completeness of the problem of finding Nash equilibriums is proved.
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References
Z. Drezner, K. Klamroth, A. Schobel, and G. Wesolowsky, “The Weber Problem,” in Facility Location: Applications and Theory (Springer, Berlin, 2004), pp. 1–36.
V. L. Beresnev, Discrete Location Problems and Polynomials of Boolean Variables (Institut matematiki, SO RAN, Novosibirsk, 2005) [in Russian].
Y. Chan, Location, Transport and Land-Use. Modelling Spatial-Temporal Information (Springer, Berlin, 2005).
V. R. Khachaturov, V. E. Veselovskii, A. V. Zlotov, et al., Combinatorial Methods and Algorithms for Solving Large-Scale Discrete Optimization Problems (Nauka, Moscow, 2000) [in Russian].
H. A. Eiselt and C.-L. Sandblom, Decision Analysis, Location Models, and Scheduling Problems (Springer, Berlin, 2004).
Discrete Location Theory, Ed. By P. B. Mirchandani, and R. L. Francis (Wiley, Chichester, 1990).
V. Beresnev, E. Kh. Gimadi, and V. T. Dement’ev, Extremal Standardization Problems (Nauka, Novosibirsk, 1978) [in Russian].
B. Korte and J. Vygen, Combinatorial Optimization: Theory and Algorithms, 3rd ed. (Springer, Berlin, 2005).
A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency (Springer, Berlin, 2003).
J. Krarup and P. M. Pruzan, “The Simple Plant Location Problem: Survey and Synthesis,” European J. Operat. Res. 12, 36–81 (1983).
E. V. Alekseeva and Yu. A. Kochetov, “Genetic Local Search for the p-Median Problem with Client’s Preferences,” Diskretnyi Analiz Issl. Operatsii, Ser. 2, 14(1), 3–31 (2007).
L. E. Gorbachevskaya, Polynomially Solvable and NP-Hard Standardization Problems, Candidate’s Dissertation in Mathematics and Physics (IM SO RAN, Novosibirsk, 1998).
C. M. C. Rodriquez and J. A. M. Perez, “Multiple Voting Location Problems,” European J. Operat. Res. 191, 437–453 (2008).
F. Plastria and L. Vanhaverbeke, “Discrete Models for Competitive Location with Foresight,” Comput. Operat. Res. 35, 683–700 (2008).
T. Drezner and H. A. Eiselt, “Consumers in Competitive Location Models,” in Facility Location: Applications and Theory (Springer, Berlin, 2004), pp. 151–178.
E. V. Alekseeva and N. A. Kochetova, “Upper and Lower Bounds for the Competitive p-Median Problem,” in Trudy XIV Baikal’skoi mezhdunarodnoi shkoly-seminara “Metody optimizatsii i ikh prilozheniya, Severobaikal’sk, 2008, Vol. 1, pp. 563–569 [in Russian].
E. V. Alekseeva and A. V. Orlov, “A Genetic Algorithm for the Competitive p-Median Problem,” in Trudy XIV Baikal’skoi mezhdunarodnoi shkoly-seminara “Metody optimizatsii i ikh prilozheniya, Severobaikal’sk,” 2008, Vol. 1, pp. 570–576 [in Russian].
E. Koutsoupias and C. Papadimitriou, “Worst-Case Equilibria,” in Proc. XVI Ann. Symposium on the Theory and Aspects of Computer Science, Trier, Germany, 1999 pp. 404–413.
A. Vetta, “Nash Equilibria in Competitive Societies, with Applications to Facility Location, Traffic Routing, and Auctions,” in Proc. XLIII Ann. IEEE Symposium on the Foundations of Computer Science, Vancouver, Canada, 2002 pp. 416–425.
Yu. A. Kochetov, M. G. Pashchenko, and A. V. Plyasunov, “On the Complexity of Local Search in the p-Median Problem,” Diskretnyi Analiz Issl. Operatsii, Ser. 2, 12(2), 44–71 (2005).
Yu. A. Kochetov, “Nash Equilibria in Game Location Models,” in Trudy XIV Baikal’skoi mezhdunarodnoi shkolyseminara “Metody optimizatsii i ikh prilozheniya, Severobaikal’sk, 2008, Vol. 1, pp. 119–127 [in Russian].
E. Tardos and T. Wexler, “Network Formation Games and the Potential Function Method” in Algorithmic Game Theory (Univ. Press, Cambridge, 2007), pp. 487–516.
Yu. A. Kochetov, “Computational Bounds for Local Search in Combinatorial Optimization,” Zh. Vychisl. Mat. Mat. Fiz. 48(5), 747–764 (2008) [Comput. Math. Math. Phys. 747–763 48, (2008)].
M. Yannakakis, “Computational Complexity,” in Local Search in Combinatorial Optimization (Wiley, Chichester, 1997), pp. 19–55.
T. Vredeveld and J. K. Lenstra, “On Local Search for the Generalized Graph Coloring Problem,” Operat. Res. Letts 31(4), 28–34 (2003).
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Original Russian Text © A.V. Kononov, Yu.A. Kochetov, A.V. Plyasunov, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, Vol. 49, No. 6, pp. 1037–1054.
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Kononov, A.V., Kochetov, Y.A. & Plyasunov, A.V. Competitive facility location models. Comput. Math. and Math. Phys. 49, 994–1009 (2009). https://doi.org/10.1134/S0965542509060086
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DOI: https://doi.org/10.1134/S0965542509060086