computational complexity

, Volume 17, Issue 3, pp 353–376 | Cite as

On the Complexity of Succinct Zero-Sum Games

  • Lance Fortnow
  • Russell Impagliazzo
  • Valentine Kabanets
  • Christopher Umans
Article

Abstract.

We study the complexity of solving succinct zero-sum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M(i,j) = C(i,j). We complement the known EXP-hardness of computing the exact value of a succinct zero-sum game by several results on approximating the value. (1) We prove that approximating the value of a succinct zero-sum game to within an additive error is complete for the class promise-\(S^{p}_{2}\), the “promise” version of \(S^{p}_{2}\). To the best of our knowledge, it is the first natural problem shown complete for this class. (2) We describe a ZPPNP algorithm for constructing approximately optimal strategies, and hence for approximating the value, of a given succinct zero-sum game. As a corollary, we obtain, in a uniform fashion, several complexity-theoretic results, e.g., a ZPPNP algorithm for learning circuits for SAT (Bshouty et al., JCSS, 1996) and a recent result by Cai (JCSS, 2007) that \(S^{p}_{2} \subseteq\)ZPPNP. (3) We observe that approximating the value of a succinct zero-sum game to within a multiplicative factor is in PSPACE, and that it cannot be in promise-\(S^{p}_{2}\) unless the polynomial-time hierarchy collapses. Thus, under a reasonable complexity-theoretic assumption, multiplicative-factor approximation of succinct zero-sum games is strictly harder than additive-error approximation.

Keywords.

Succinct zero-sum games approximating the value of a zero-sum game completeness \(S^{p}_{2}\) ZPPNP 

Subject classification.

68Q15 68Q17 68Q32 03D15 91A05 

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Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  • Lance Fortnow
    • 1
  • Russell Impagliazzo
    • 2
  • Valentine Kabanets
    • 3
  • Christopher Umans
    • 4
  1. 1.Department of Computer ScienceUniversity of ChicagoChicagoUSA
  2. 2.Department of Computer ScienceUniversity of California – San DiegoLa JollaUSA
  3. 3.School of Computing ScienceSimon Fraser UniversityVancouverCanada
  4. 4.Department of Computer ScienceCalifornia Institute of TechnologyPasadenaUSA

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