On a Class of Optimization Problems with No “Efficiently Computable” Solution
- 26 Downloads
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.
Keywords
Optimal Strategy Pure Strategy Consistent Equation Checkable Proof Inconsistent EquationPreview
Unable to display preview. Download preview PDF.
References
- 1.D. G. Kabe, “Laws of large numbers for random payoff games,” Indust. Math., 33, 73–87 (1983).MathSciNetMATHGoogle Scholar
- 2.J. Jonasson, “On the optimal strategy in a random game,” Electron. Comm. Probab., 9, 132–139 (2004).MathSciNetCrossRefMATHGoogle Scholar
- 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.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.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.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.J. Håstad, “Some optimal inapproximability results,” in: Proceedings of 29th Annual Symposium on Theory of Computation (1997), pp. 1–10.Google Scholar
- 8.Y. Luke, Mathematical Functions and Their Approximations, Academic Press (1975).Google Scholar
- 9.B. D. McKay, “On Littlewood’s estimate for the binomial distribution,” Adv. Appl. Probab., 21 475–478 (1989).MathSciNetCrossRefMATHGoogle Scholar
- 10.J. E. Littlewood, “On the probability in the tail of a binomial distribution,” Adv. Appl. Prob., 1, 43–52 (1969).MathSciNetCrossRefMATHGoogle Scholar
- 11.B. Bollobas, Random Graphs, 2nd edition, Cambridge Univ. Press, Cambridge (2001).Google Scholar
- 12.S. Arora and B. Barak, Computational Complexity: A Modern Approach, Cambridge Univ. Press, Cambridge (2009).Google Scholar
- 13.I. Dinur, “The PCP theorem by gap amplification,” Technical Report TR05-046, ECCC (2005), Revision 1, available from http://eccc.uni-trier.de/eccc-reports/2005/TR05-046/.