Some Remarks on the Incompressibility of Width-Parameterized SAT Instances

  • Bangsheng Tang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7285)

Abstract

Compressibility of a formula regards reducing the length of the input, or some other parameter, while preserving the solution. Any 3-SAT instance on N variables can be represented by O(N 3) bits; [4] proved that the instance length in general cannot be compressed to O(N 3 − ε ) bits under the assumption \(\mathbf{NP}\not\subseteq\mathbf{coNP}\) /poly, which implies that the polynomial hierarchy does not collapse. This note initiates research on compressibility of SAT instances parameterized by width parameters, such as tree-width or path-width. Let SAT tw (w(n)) be the satisfiability instances of length n that are given together with a tree-decomposition of width O(w(n)), and similarly let SAT pw (w(n)) be instances with a path-decomposition of width O(w(n)). Applying simple techniques and observations, we prove conditional incompressibility for both instance length and width parameters: (i) under the exponential time hypothesis, given an instance φ of SAT tw (w(n)) it is impossible to find within polynomial time a φ′ that is satisfiable if and only if φ is satisfiable and tree-width of φ′ is half of φ; and (ii) assuming a scaled version of \(\mathbf{NP}\not\subseteq\mathbf{coNP}\) /poly, any 3-SAT pw (w(n)) instance of N variables cannot be compressed to O(N 1 − ε ) bits.

Keywords

Conjunctive Normal Form Tree Decomposition Width Parameter Input Instance Input Length 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Bangsheng Tang
    • 1
  1. 1.Institute for Interdisciplinary Information SciencesTsinghua UniversityBeijingChina

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