Some Remarks on the Incompressibility of Width-Parameterized SAT Instances

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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−e ) 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−e ) bits.