Journal of Mathematical Sciences

, Volume 215, Issue 6, pp 706–714 | Cite as

On a Class of Optimization Problems with No “Efficiently Computable” Solution


It is well known that large random structures may have nonrandom macroscopic properties. We give an example of nonrandom properties for a class of large optimization problems related to the computational problem MAXFLS= of calculating the maximum number of consistent equations in a given overdetermined system of linear equations. A problem of this kind is faced by a decision maker (an Agent) choosing means to protect a house from natural disasters. For this class we establish the following. There is no “efficiently computable” optimal strategy of the Agent. As the size of a random instance of the optimization problem goes to infinity, the probability that the uniform mixed strategy of the Agent is ε-optimal goes to one. Moreover, there is no “efficiently computable” strategy of the Agent that is substantially better for each instance of the optimization problem. Bibliography: 13 titles.


Optimal Strategy Pure Strategy Consistent Equation Checkable Proof Inconsistent Equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. G. Kabe, “Laws of large numbers for random payoff games,” Indust. Math., 33, 73–87 (1983).MathSciNetMATHGoogle Scholar
  2. 2.
    J. Jonasson, “On the optimal strategy in a random game,” Electron. Comm. Probab., 9, 132–139 (2004).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    A. McLennan and J. Berg, “Asymptotic expected number of Nash equilibria of two-player normal form games,” Games Econom. Behav., 51, No. 2, 264–295 (2005).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    A. Tanikawa, “On stability of the values of random zero-sum games,” Internat. J. Innov. Comput. Inform. Control, 7, No. 1, 133–140 (2011).Google Scholar
  5. 5.
    F. Sandomirskiy, “On typical properties of large zero-sum games,” in: L. Petrosjan and N. Zenkevich (eds.), The International Conference SING-GTM 2015, Abstracts, St.Petersburg (2015).Google Scholar
  6. 6.
    E. Amaldi and V. Kann, “The complexity and approximability of finding maximum feasible subsystems of linear relations,” Theoret. Comput. Sci., 147, 181–210 (1995).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    J. Håstad, “Some optimal inapproximability results,” in: Proceedings of 29th Annual Symposium on Theory of Computation (1997), pp. 1–10.Google Scholar
  8. 8.
    Y. Luke, Mathematical Functions and Their Approximations, Academic Press (1975).Google Scholar
  9. 9.
    B. D. McKay, “On Littlewood’s estimate for the binomial distribution,” Adv. Appl. Probab., 21 475–478 (1989).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    J. E. Littlewood, “On the probability in the tail of a binomial distribution,” Adv. Appl. Prob., 1, 43–52 (1969).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    B. Bollobas, Random Graphs, 2nd edition, Cambridge Univ. Press, Cambridge (2001).Google Scholar
  12. 12.
    S. Arora and B. Barak, Computational Complexity: A Modern Approach, Cambridge Univ. Press, Cambridge (2009).Google Scholar
  13. 13.
    I. Dinur, “The PCP theorem by gap amplification,” Technical Report TR05-046, ECCC (2005), Revision 1, available from

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsSt.PetersburgRussia

Personalised recommendations