Some Remarks on the Incompressibility of Width-Parameterized SAT Instances

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.