Abstract
The bin packing problem, a classical problem in combinatorial optimization, has recently been studied from the viewpoint of algorithmic game theory. In this bin packing game each item is controlled by a selfish player minimizing its personal cost, which in this context is defined as the relative contribution of the size of the item to the total load in the bin.
We introduce a related game, the so-called bin coloring game, in which players control colored items and each player aims at packing its item into a bin with as few different colors as possible.
We establish existence of Nash and strong as well as weakly and strictly Pareto optimal equilibria in these games in the cases of capacitated and uncapacitated bins. For both kinds of games we determine the prices of anarchy and stability concerning those four equilibrium concepts. Furthermore, we show that extreme Nash equilibria, representatives of the set of Nash equilibria with minimal or maximal number of colors in a bin, can be found in time polynomial in the number of items for the uncapacitated case.
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References
Andelman N, Feldman M, Mansour Y (2009) Strong price of anarchy. Games Econ Behav 65(2):289–317
Anshelevich E, Dasgupta A, Tardós E, Wexler T (2008) Near-optimal network design with selfish agents. Theory Comput 4:77–109
Aumann RJ (1959) Acceptable points in general cooperative n-person games. In: Luce RD, Tucker AW (eds) Contributions to the theory of games, vol IV. Princeton University Press, Princeton, pp 287–324
Bilò V (2006) On the packing of selfish items. In: Proceedings of the 20th international parallel symposium on distributed processing (IPDPS). IEEE Press, New York, pp 25–29
Chien S, Sinclair A (2009) Strong and Pareto price of anarchy in congestion games. In: Albers S, Marchetti-Spaccamela A, Matias Y, Nikoletseas S, Thomas W (eds) Proceedings of the 36th international colloquium on automata, languages and programming (ICALP). LNCS, vol 5555. Springer, Berlin, pp 279–291
Coffman EG Jr, Csirik J (2007) Performance guarantees for one-dimensional bin packing. In: Gonzales TF (ed) Handbook of approximation algorithms and metaheuristics. Chapman & Hall, London. Chap 32
Csirik J, Leung JYT (2007) Variants of classical one-dimensional bin packing. In: Gonzalez TF (ed) Handbook of approximation algorithms and metaheuristics. Chapman & Hall, London. Chap 33
Csirik J, Woeginger GJ (1998) On-line packing and covering problems. In: Fiat A, Woeginger GJ (eds) Online algorithms: the state of the art. LNCS, vol 1442. Springer, Berlin, pp 147–177. Chap 7
Czumaj A, Vöcking B (2007) Tight bounds for worst-case equilibria. ACM Trans Algorithms 3(1):4
Epstein L, Kleiman E (2009) Selfish bin packing. Algorithmica. doi:10.1007/s00453-009-9348-6
Epstein A, Feldman M, Mansour Y (2009) Efficient graph topologies in network routing games. Games Econ Behav 66(1):115–125
Epstein L, Favrholdt LM, Kohrt JS (2009) Comparing online algorithms for bin packing problems. J Sched. doi:10.1007/s10951-009-0129-5
Feldmann R, Gairing M, Lücking T, Monien B, Rode M (2003) Nashification and the coordination ratio for a selfish routing game. In: Baeten JCM, Lenstra JK, Parrow J, Woeginger GJ (eds) Proceedings of the 30th international colloquium on automata, languages and programming (ICALP). LNCS, vol 2719. Springer, Berlin, pp 514–526
Fischer S, Vöcking B (2007) On the structure and complexity of worst-case equilibria. Theor Comput Sci 378(2):165–174
Fotakis D, Kontogiannis S, Koutsoupias E, Mavronicolas M, Spirakis P (2009) The structure and complexity of Nash equilibria for a selfish routing game. Theor Comput Sci 410(36):3305–3326
Gassner E, Hatzl J, Krumke SO, Sperber H, Woeginger GJ (2009) How hard is it to find extreme Nash equilibria in network congestion games? Theor Comput Sci 410(47–49):4989–4999. doi:10.1016/j.tcs.2009.07.046
Hiller B, Vredefeld T (2008) Probabilistic analysis of online bin coloring algorithms via stochastic comparison. In: Halperin D, Mehlhorn K (eds) Proceedings of the 16th annual European symposium on algorithms (ESA). LNCS, vol 5193. Springer, Berlin, pp 528–539
Holzman R, Law-Yone N (1997) Strong equilibrium in congestion games. Games Econ Behav 21(1–2):85–101
Kamin N (1998) On-line optimization of order picking in an automated warehouse. Ph.D. thesis, Technische Universität Berlin
Koutsoupias E, Papadimitriou C (1999) Worst-case equilibria. In: Meinel C, Tison S (eds) Proceedings of the 16th international symposium on theoretical aspects of computer science (STACS). LNCS, vol 1563. Springer, Berlin, pp 404–413
Koutsoupias E, Papadimitriou C (2009) Worst-case equilibria. Comput Sci Rev 3:65–69
Krumke SO, de Paepe WE, Stougie L, Rambau J (2008) Bincoloring. Theor Comput Sci 407(1–3):231–241
Lin M, Lin Z, Xu J (2008) Almost optimal solutions for bin coloring problems. J Comb Optim 16(1):16–27
Mas-Colell A, Whinston MD, Green JG (1995) Microeconomic theory. Oxford University Press, London
Mavronicolas M, Spirakis P (2007) The price of selfish routing. Algorithmica 48(1):91–126
Pigou AC (1920) The economics of welfare. Macmillan Co, New York
Schmähler P (2010) Bin coloring games. Master thesis (teaching qualification), University of Kaiserslautern
Shachnai H, Tamir T (2001) Polynomial time approximation schemes for class-constrained packing problems. J Sched 4(6):313–338
Shachnai H, Tamir T (2004) Tight bounds for online class-constrained packing. Theor Comput Sci 321(1):103–123
Xavier EC, Miyazawa FK (2008) The class constrained bin packing problem with applications to video-on-demand. Theor Comput Sci 393(1–3):240–259
Yu G, Zhang G (2008) Bin packing of selfish items. In: Papadimitriou C, Zhang S (eds) Proceedings of the 4th international workshop on Internet and network economics (WINE). LNCS, vol 5385. Springer, Berlin, pp 446–453
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Epstein, L., Krumke, S.O., Levin, A. et al. Selfish bin coloring. J Comb Optim 22, 531–548 (2011). https://doi.org/10.1007/s10878-010-9302-1
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DOI: https://doi.org/10.1007/s10878-010-9302-1