Algorithmic Game Theory: Some Greatest Hits and Future Directions

  • Tim Roughgarden
Conference paper
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 273)


We give a brief and biased survey of the past, present, and future of research on the interface of theoretical computer science and game theory.


Nash Equilibrium Allocation Algorithm Electronic Commerce Combinatorial Auction Congestion Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    H. Ackermann, H. Röglin, and B. Vöcking. On the impact of combinatorial structure on congestion games. In FOCS ’06, pages 613–622.Google Scholar
  2. 2.
    H. Ackermann and A. Skopalik. On the complexity of pure Nash equilibria in player-specific network congestion games. In WINE ’07, pages 419–430.Google Scholar
  3. 3.
    G. Aggarwal, A. Fiat, A. V. Goldberg, J. D. Hartline, N. Immorlica, and M. Sudan. Derandomization of auctions. In STOC ’05, pages 619–625.Google Scholar
  4. 4.
    E. Altman, T. Boulogne, R. El Azouzi, T. Jiménez, and L. Wynter. A survey on networking games in telecommunications. Computers & Operations Research, 33(2):286–311, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    N. Andelman, M. Feldman, and Y. Mansour. Strong price of anarchy. In SODA ’07, pages 189–198.Google Scholar
  6. 6.
    E. Anshelevich, A. Dasgupta, J. Kleinberg, É. Tardos, T. Wexler, and T. Roughgarden. The price of stability for network design with fair cost allocation. In FOCS ’04, pages 295–304.Google Scholar
  7. 7.
    A. Archer. Mechanisms for Discrete Optimization with Rational Agents. PhD thesis, Cornell University, 2004.Google Scholar
  8. 8.
    A. Archer and R. D. Kleinberg. Truthful germs are contagious: a local-to-global characterization of truthfulness. In EC ’08.Google Scholar
  9. 9.
    A. Archer, C. H. Papadimitriou, K. Talwar, and É. Tardos. An approximate truthful mechanism for combinatorial auctions with single parameter agents. In SODA ’03, pages 205–214.Google Scholar
  10. 10.
    A. Archer and É. Tardos. Truthful mechanisms for one-parameter agents. In FOCS ’01, pages 482–491.Google Scholar
  11. 11.
    K. Arrow and G. Debreu. Existence of an equilibrium for a competitive economy. Econometrica, 22:265–290, 1954.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    R. J. Aumann. Subjectivity and correlation in randomized strategies. Journal of Mathitatical Economics, 1(1):67–96, 1974.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    B. Awerbuch, Y. Azar, and Y. Bartal. On-line generalized Steiner problit. In SODA ’96, pages 68–74.Google Scholar
  14. 14.
    B. Awerbuch, Y. Azar, A. Epstein, V. S. Mirrokni, and A. Skopalik. Fast convergence to nearly optimal solutions in potential games. In EC ’08.Google Scholar
  15. 15.
    M. Babaioff, R. D. Kleinberg, and C. H. Papadimitriou. Congestion games with malicious players. In EC ’07, pages 103–112.Google Scholar
  16. 16.
    M. Babaioff, R. Lavi, and E. Pavlov. Single-value combinatorial auctions and implitentation in undominated strategies. In SODA ’06, pages 1054–1063.Google Scholar
  17. 17.
    Y. Bartal, R. Gonen, and N. Nisan. Incentive compatible multi unit combinatorial auctions. In TARK ’03, pages 72–87.Google Scholar
  18. 18.
    M. J. Beckmann, C. B. McGuire, and C. B. Winsten. Studies in the Economics of Transportation. Yale University Press, 1956.Google Scholar
  19. 19.
    D. P. Bertsekas and R. G. Gallager. Data Networks. Prentice-Hall, 1987. Second Edition, 1991.Google Scholar
  20. 20.
    S. Bikhchandani, S. Chatterji, R. Lavi, A. Mu’alem, N. Nisan, and A. Sen. Weak monotonicity characterizes dominant strategy implitentation. Econometrica, 74(4): 1109–1132, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    A. Blum, E. Even-Dar, and K. Ligett. Routing without regret: On convergence to Nash equilibria of regret-minimizing algorithms in routing games. In PODC ’06, pages 45–52.Google Scholar
  22. 22.
    A. Blum, M. Hajiaghayi K. Ligett, and A. Roth. Regret minimization and the price of total anarchy. In STOC ’08.Google Scholar
  23. 23.
    A. Blum and Y. Mansour. Learning, regret minimization, and equilibria. In Nisan et al. [94], chapter 4, pages 79–101.Google Scholar
  24. 24.
    L. Blumrosen and N. Nisan. Combinatorial auctions. In Nisan et al. [94], chapter 11, pages 267–299.Google Scholar
  25. 25.
    C. Borgs, J. Chayes, N. Immorlica, A. T. Kalai, V. S. Mirrokni, and C. H. Papadimitriou. The myth of the folk theorit. In STOC ’08.Google Scholar
  26. 26.
    H. Bosse, J. Byrka, and E. Markakis. New algorithms for approximate Nash equilibria in bimatrix games. In WINE ’07, pages 17–29.Google Scholar
  27. 27.
    H. Chen, T. Roughgarden, and G. Valiant. Designing networks with good equilibria. In SODA ’08, pages 854–863.Google Scholar
  28. 28.
    P.-A. Chen and D. Kitpe. Altruism, selfishness, and spite in traffic routing. In EC ’08.Google Scholar
  29. 29.
    X. Chen, X. Deng, and S.-H. Teng. Settling the complexity of two-player Nash equilibria. Journal of the ACM, 2008.Google Scholar
  30. 30.
    S. Chien and A. Sinclair. Convergence to approximate Nash equilibria in congestion games. In SODA ’07, pages 169–178.Google Scholar
  31. 31.
    G. Christodoulou and E. Koutsoupias. On the price of anarchy and stability of correlated equilibria of linear congestion games. In EC ’05, pages 59–70.Google Scholar
  32. 32.
    G. Christodoulou, E. Koutsoupias, and A. Nanavati. Coordination mechanisms. In ICALP ’04, pages 345–357.Google Scholar
  33. 33.
    G. Christodoulou, E. Koutsoupias, and A. Vidali. A characterization of 2-player mechanisms for scheduling. Submitted, 2008.Google Scholar
  34. 34.
    B. Codenotti and K. Varadarajan. Computation of market equilibria by convex programming. In Nisan et al. [94], chapter 6, pages 135–158.Google Scholar
  35. 35.
    R. Cominetti, J. R. Correa, and N. E. Stier Moses. The impact of oligopolistic competition in networks. 2008.Google Scholar
  36. 36.
    A. Condon. The complexity of stochastic games. Information and Computation, 96:203–224, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    P. Cramton. Spectrum auctions. In Handbook of Telecommunications Economics, chapter 14, pages 605–639. 2002.Google Scholar
  38. 38.
    P. Cramton, Y. Shoham, and R. Steinberg, editors. Combinatorial Auctions. MIT Press, 2006.Google Scholar
  39. 39.
    C. Daskalakis, P. W. Goldberg, and C. H. Papadimitriou. The complexity of comuting a Nash equilibria. SIAM Journal on Computing, 2008.Google Scholar
  40. 40.
    P. Dhangwatnotai, S. Dobzinski, S. Dughmi, and T. Roughgarden. Truthful approximation schites for single-parameter agents. Submitted, 2008.Google Scholar
  41. 41.
    S. Dobzinski and N. Nisan. Limitations of VCG-based mechanisms. In STOC ’07, pages 338–344.Google Scholar
  42. 42.
    S. Dobzinski and M. Sundararajan. On characterizations of truthful mechanisms for combinatorial auctions and scheduling. In EC ’08.Google Scholar
  43. 43.
    L. Epstein and J. Sgall. Approximation schites for scheduling on uniformly related and identical parallel machines. Algorithmica, 39(1):43–57, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    E. Even-Dar, A. Kesselman, and Y. Mansour. Convergence time to Nash equilibria. In ICALP ’03, pages 502–513.Google Scholar
  45. 45.
    A. Fabrikant and C. H. Papadimitriou. The complexity of game dynamics: BGP oscillations, sink equlibria, and beyond. In SODA ’08, pages 844–853.Google Scholar
  46. 46.
    A. Fabrikant, C. H. Papadimitriou, and K. Talwar. The complexity of pure Nash equilibria. In STOC ’04, pages 604–612.Google Scholar
  47. 47.
    U. Feige. On maximizing welfare when utility functions are subadditive. In STOC ’06, pages 41–50.Google Scholar
  48. 48.
    J. Feigenbaum, M. Schapira, and S. Shenker. Distributed algorithmic mechanism design. In Nisan et al. [94], chapter 14, pages 363–384.Google Scholar
  49. 49.
    A. Fiat, H. Kaplan, M. Levy, S. Olonetsky, and R. Shabo. On the price of stability for designing undirected networks with fair cost allocations. In ICALP ’06, pages 608–618.Google Scholar
  50. 50.
    S. Fischer and B. Vöcking. On the evolution of selfish routing. In ESA ’04, pages 323–334.Google Scholar
  51. 51.
    E. J. Friedman and S. J. Shenker. Learning and implitentation on the Internet. Working paper, 1997.Google Scholar
  52. 52.
    A. Gilpin, T. Sandholm, and T. B. Sorensen. Potential-aware automated abstraction of sequential games, and holistic equilibrium analysis of Texas Hold’it poker. In AAAI ’07.Google Scholar
  53. 53.
    M. X. Goitans, V. Mirrokni, and A. Vetta. Sink equilibria and convergence. In FOCS ’05, pages 142–151.Google Scholar
  54. 54.
    J. Y. Halpern. Computer science and game theory: A brief survey. In S. N. Durlauf and L. E. Blume, editors, Palgrave Dictionary of Economics. 2008.Google Scholar
  55. 55.
    S. Hart and Y. Mansour. The communication complexity of uncoupled Nash equilibrium procedures. Games and Economic Behavior, 2008.Google Scholar
  56. 56.
    J. Hartline and A. Karlin. Profit maximization in mechanism design. In Nisan et al. [94], chapter 13, pages 331–362.Google Scholar
  57. 57.
    J. D. Hartline and T. Roughgarden. Optimal mechanism design and money burning. In STOC ’08.Google Scholar
  58. 58.
    D. Hochbaum and D. B. Shmoys. A polynomial approximation schite for scheduling on uniform processors: Using the dual approximation approach. SIAM J. Comput., 17(3):539–551, 1988.MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    M. Imase and B. M. Waxman. Dynamic Steiner tree problit. SIAM Journal on Discrete Mathitatics, 4(3), 1991.zbMATHGoogle Scholar
  60. 60.
    M. O. Jackson. A crash course in implitentation theory. Social Choice and Welfare, 18(4):655–708, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    K. Jain and M. Mahdian. Cost sharing. In Nisan et al. [94], chapter 15, pages 385–410.Google Scholar
  62. 62.
    R. Johari. The price of anarchy and the design of scalable resource allocation mechanisms. In Nisan et al. [94], chapter 21, pages 543–568.Google Scholar
  63. 63.
    D. S. Johnson. The NP-completeness column: Finding needles in haystacks. ACM Transactions on Algorithms, 3(2), 2007. Article 24.MathSciNetCrossRefGoogle Scholar
  64. 64.
    D. S. Johnson, C. H. Papadimitriou, and M. Yannakakis. How easy is local search? Journal of Computer and Systit Sciences, 37(1):79–100, 1988.MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    E. Kalai and D. Samet. On weighted Shapley values. International Journal of Game Theory, 16(3):205–222, 1987.MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    G. Karakostas and A. Viglas. Equilibria for networks with malicious users. Mathitatical Programming, 110(3):591–613, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    D. Koller and A. Pfeffer. Representations and solutions for game-theoretic problits. Artificial Intelligence, 94(1-2):167–215, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    S. C. Kontogiannis and P. G. Spirakis. Atomic selfish routing in networks: A survey. In WINE ’05, pages 989–1002.Google Scholar
  69. 69.
    E. Koutsoupias and C. H. Papadimitriou. Worst-case equilibria. In STACS ’99, pages 404–413.Google Scholar
  70. 70.
    A. Kovács. Tighter approximation bounds for LPT scheduling in two special cases. In CIAC, pages 187–198, 2006.Google Scholar
  71. 71.
    S. Lahaie, D. Pennock, A. Saberi, and R. Vohra. Sponsored search auctions. In Nisan et al. [94], chapter 28.Google Scholar
  72. 72.
    R. Lavi. Computationally efficient approximation mechanisms. In Nisan et al. [94], chapter 12, pages 301–329.Google Scholar
  73. 73.
    R. Lavi, A. Mu’alem, and N. Nisan. Towards a characterization of truthful combinatorial auctions. In FOCS ’03, pages 574–583.Google Scholar
  74. 74.
    R. Lavi and C. Swamy. Truthful mechanism design for multi-dimensional scheduling via cycle monotonicity. In EC ’07, pages 252–261.Google Scholar
  75. 75.
    B. Lehmann, D. J. Lehmann, and N. Nisan. Combinatorial auctions with decreasing marginal utilities. In EC ’01, pages 18–28.Google Scholar
  76. 76.
    D. Lehmann, L. I. O’Callaghan, and Y. Shoham. Truth revelation in approximately efficient combinatorial auctions. Journal of the ACM, 49(5):577–602, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    C. E. Litke and J. T. Howson, Jr. Equilibrium points of bimatrix games. SIAM Journal, 12(2):413–423, 1964.MathSciNetzbMATHGoogle Scholar
  78. 78.
    H. Levin, M. Schapira, and A. Zohar. Interdomain routing and games. In STOC ’08, 2008.Google Scholar
  79. 79.
    N. Linial. Game-theoretic aspects of computing. In R. J. Aumann and S. Hart, editors, Handbook of Game Theory with Economic Applications, volume 2, chapter 38, pages 1339–1395. 1994.CrossRefzbMATHGoogle Scholar
  80. 80.
    R. J. Lipton, E. Markakis, and A. Mehta. Playing large games using simple strategies. In EC ’03, pages 36–41.Google Scholar
  81. 81.
    M. L. Littman and P. Stone. A polynomial-time Nash equilibrium algorithm for repeated games. Decision Support Systits, 39(1):55–66, 2005.CrossRefGoogle Scholar
  82. 82.
    N. Megiddo and C. H. Papadimitriou. On total functions, existence theorits and computational complexity. Theoretical Computer Science, 81(2):317–324, 1991.MathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    A. Mehta, T. Roughgarden, and M. Sundararajan. Beyond Moulin mechanisms. In Proceedings of the 8th ACM Conference on Electronic Commerce (EC), pages 1–10, 2007.Google Scholar
  84. 84.
    M. Mihail, C. H. Papadimitriou, and A. Saberi. On certain connectivity properties of the Internet topology. Journal of Computer and Systit Sciences, 72(2):239–251, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    P. B. Miltersen and T. B. Sørensen. Fast algorithms for finding proper strategies in game trees. In SODA ’08, pages 874–883.Google Scholar
  86. 86.
    V. S. Mirrokni and A. Vetta. Convergence issues in competitive games. In APPROX ’04, pages 183–194.Google Scholar
  87. 87.
    D. Monderer. Monotonicity and implitentability. In EC ’08.Google Scholar
  88. 88.
    D. Monderer and L. S. Shapley. Potential games. Games and Economic Behavior, 14(1):124–143, 1996.MathSciNetCrossRefzbMATHGoogle Scholar
  89. 89.
    A. Mu’alem and N. Nisan. Truthful approximation mechanisms for restricted combinatorial auctions. In AAAI ’02, pages 379–384.Google Scholar
  90. 90.
    R. Myerson. Optimal auction design. Mathitatics of Operations Research, 6:58–73, 1981.MathSciNetCrossRefzbMATHGoogle Scholar
  91. 91.
    J. F. Nash. Equilibrium points in N-person games. Proceedings of the National Acadity of Science, 36(1):48–49, 1950.MathSciNetCrossRefzbMATHGoogle Scholar
  92. 92.
    N. Nisan. Introduction to mechanism design (for computer scientists). In Nisan et al. [94], chapter 9, pages 209–241.Google Scholar
  93. 93.
    N. Nisan and A. Ronen. Algorithmic mechanism design. Games and Economic Behavior, 35(1/2):166–196, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  94. 94.
    N. Nisan, T. Roughgarden, É. Tardos, and V. Vazirani, editors. Algorithmic Game Theory. Cambridge University Press, 2007.Google Scholar
  95. 95.
    N. Nisan, M. Schapira, and A. Zohar. Best-reply mechanisms. Working paper, 2008.Google Scholar
  96. 96.
    M. J. Osborne and A. Rubinstein. A Course in Game Theory. MIT Press, 1994.Google Scholar
  97. 97.
    C. H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and Systit Sciences, 48:498–532, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  98. 98.
    C. H. Papadimitriou. The complexity of finding nash equilibria. In Nisan et al. [94], chapter 2, pages 29–51.Google Scholar
  99. 99.
    C. H. Papadimitriou and T. Roughgarden. Computing correlated equilibria in multi-player games. Journal of the ACM, 2008.Google Scholar
  100. 100.
    C. H. Papadimitriou, M. Schapira, and Y. Singer. On the hardness of being truthful. Manuscript, 2008.Google Scholar
  101. 101.
    D. Parkes. Iterative Combinatorial Auctions: Achieving Economic and Computational Efficiency. PhD thesis, University of Pennsylvania, 2001.Google Scholar
  102. 102.
    D. C. Parkes. Online mechanisms. In Nisan et al. [94], chapter 16, pages 411–439.Google Scholar
  103. 103.
    A. C. Pigou. The Economics of Welfare. Macmillan, 1920.Google Scholar
  104. 104.
    M. O. Rabin. Effective computability of winning strategies. In M. Dresher, A. W. Tucker, and P. Wolfe, editors, Contributions to the Theory Games, volume 3. Princeton University Press, 1957.Google Scholar
  105. 105.
    J. Riley and W. Samuelson. Optimal auctions. American Economic Review, 71:381–92, 1981.Google Scholar
  106. 106.
    K. Roberts. The characterization of implitentable choice rules. In J.-J. Laffont, editor, Aggregation and Revelation of Preferences, pages 321–349. 1979.Google Scholar
  107. 107.
    J. C. Rochet. A necessary and sufficient condition for rationalizability in a quasilinear context. Journal of Mathitatical Economics, 16:191–200, 1987.MathSciNetCrossRefzbMATHGoogle Scholar
  108. 108.
    R. W. Rosenthal. A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory, 2(1):65–67, 1973.MathSciNetCrossRefzbMATHGoogle Scholar
  109. 109.
    T. Roughgarden. The price of anarchy is independent of the network topology. Journal of Computer and Systit Sciences, 67(2):341–364, 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  110. 110.
    T. Roughgarden. Selfish Routing and the Price of Anarchy. MIT Press, 2005.Google Scholar
  111. 111.
    T. Roughgarden. Potential functions and the inefficiency of equilibria. In Proceedings of the International Congress of Mathitaticians, volume III, pages 1071–1094, 2006.MathSciNetzbMATHGoogle Scholar
  112. 112.
    T. Roughgarden. Routing games. In Nisan et al. [94], chapter 18, pages 461–486.Google Scholar
  113. 113.
    T. Roughgarden. Selfish routing and the price of anarchy. OPTIMA, 74:1–15, 2007.Google Scholar
  114. 114.
    T. Roughgarden. Computing equilibria: A computational complexity perspective. Economic Theory, 2008.Google Scholar
  115. 115.
    T. Roughgarden and É. Tardos. How bad is selfish routing? Journal of the ACM, 49(2):236–259, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  116. 116.
    T. Roughgarden and É. Tardos. Bounding the inefficiency of equilibria in nonatomic congestion games. Games and Economic Behavior, 49(2):389–403, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  117. 117.
    T. Roughgarden and É. Tardos. Introduction to the inefficiency of equilibria. In Nisan et al. [94], chapter 17, pages 443–459.Google Scholar
  118. 118.
    M. E. Saks and L. Yu. Weak monotonicity suffices for truthfulness on convex domains. In EC ’05, pages 286–293.Google Scholar
  119. 119.
    T. Sandholm. Algorithm for optimal winner determination in combinatorial auctions. Artificial Intelligence, 135(1):1–54, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  120. 120.
    R. Savani and B. von Stengel. Hard-to-solve bimatrix games. Econometrica, 74(2):397–429, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  121. 121.
    A. A. Schäffer and M. Yannakakis. Simple local search problits that are hard to solve. SIAM Journal on Computing, 20(1):56–87, 1991.MathSciNetCrossRefzbMATHGoogle Scholar
  122. 122.
    J. Schummer and R. V. Vohra. Mechanism design without money. In Nisan et al. [94], chapter 10, pages 243–265.Google Scholar
  123. 123.
    Y. Shoham. Computer science and game theory. Communications of the ACM, 2008.Google Scholar
  124. 124.
    Y. Shoham and K. Leyton-Brown. Multiagent Systits: Algorithmic, Game Theoretic and Logical Foundations. Cambridge University Press, 2008.Google Scholar
  125. 125.
    M. Sipser. On relativization and the existence of complete sets. In ICALP ’82, pages 523–531.Google Scholar
  126. 126.
    A. Skopalik and B. Vöcking. Inapproximability of pure Nash equilibria. In STOC ’08.Google Scholar
  127. 127.
    É. Tardos and T. Wexler. Network formation games and the potential function method. In Nisan et al. [94], chapter 19, pages 487–516.Google Scholar
  128. 128.
    H. Tsaknakis and P. G. Spirakis. An optimization approach for approximate Nash equilibria. In WINE ’07, pages 42–56.Google Scholar
  129. 129.
    V. V. Vazirani. Combinatorial algorithms for market equilibria. In Nisan et al. [94], chapter 5, pages 103–134.Google Scholar
  130. 130.
    W. Vickrey. Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16(1):8–37, 1961.MathSciNetCrossRefGoogle Scholar
  131. 131.
    B. Vöcking. Selfish load balancing. In Nisan et al. [94], chapter 20, pages 517–542.Google Scholar
  132. 132.
    R. Vohra. Paths, cycles and mechanism design. Working paper, 2007.Google Scholar
  133. 133.
    B. von Stengel. Computating equilibria for two-person games. In R. J. Aumann and S. Hart, editors, Handbook of Game Theory with Economic Applications, volume 3, chapter 45, pages 1723–1759. North-Holland, 2002.Google Scholar
  134. 134.
    B. von Stengel. Equilibrium computation for two-player games in strategic and extensive form. In Nisan et al. [94], chapter 3, pages 53–78.Google Scholar
  135. 135.
    J. Vondrak. Optimal approximation for the submodular welfare problit in the value oracle model. In STOC ’08.Google Scholar
  136. 136.
    J. G. Wardrop. Some theoretical aspects of road traffic research. In Proceedings of the Institute of Civil Engineers, Pt. II, volume 1, pages 325–378, 1952.CrossRefGoogle Scholar
  137. 137.
    M. Yannakakis. Computational complexity. In E. Aarts and J. K. Lenstra, editors, Local Search in Combinatorial Optimization, chapter 2, pages 19–55. 1997.Google Scholar

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© IFIP International Federation for Information Processing 2008

Authors and Affiliations

  • Tim Roughgarden
    • 1
  1. 1.Department of Computer ScienceStanford UniversityStanfordUSA

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