A direct construction of polynomial-size OBDD proof of pigeon hole problem

  • Authors:

  • Wěi Chén;Wenhui Zhang

  • Affiliations:

  • State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China and School of Information Science and Engineering, Graduate University of the Chinese A ...;State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China

  • Venue:
  • Information Processing Letters
  • Year:
  • 2009

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Abstract

Viewing OBDD from the explicit perspective of a propositional proof system is first proposed and studied in [A. Atserias, P.G. Kolaitis, M.Y. Vardi, Constraint propagation as a proof system, in: CP, 2004, pp. 77-91]. It has been shown that OBDD proof system defined in [A. Atserias, P.G. Kolaitis, M.Y. Vardi, Constraint propagation as a proof system, in: CP, 2004, pp. 77-91] is strictly stronger than resolution and can polynomially simulate cutting plane proof system with small coefficients CP^*. It is already shown in [W. Cook, C.R. Coullard, G. Turan, On the complexity of cutting-plane proofs, Discrete Appl. Math. 18 (1) (1987) 25-38] that there exists polynomial-size proof for pigeon hole problem PHP"n^n^+^1 of cutting plane proof system. Then it follows directly that there exists polynomial-size proof for PHP"n^n^+^1 of OBDD proof system. However, this is an indirect result. Atserias et al. [A. Atserias, P.G. Kolaitis, M.Y. Vardi, Constraint propagation as a proof system, in: CP, 2004, pp. 77-91] call for the need of a direct construction. Hereby we present such construction. Moreover, in this construction we do not need the weakening rule introduced in [A. Atserias, P.G. Kolaitis, M.Y. Vardi, Constraint propagation as a proof system, in: CP, 2004, pp. 77-91]. We believe this may shed some light on the understanding of the role of the weakening rule.