A partial k-arboretum of graphs with bounded treewidth
Theoretical Computer Science
Which problems have strongly exponential complexity?
Journal of Computer and System Sciences
Satisfiability, Branch-Width and Tseitin Tautologies
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Counting truth assignments of formulas of bounded tree-width or clique-width
Discrete Applied Mathematics
A note on width-parameterized SAT: An exact machine-model characterization
Information Processing Letters
Complexity and algorithms for well-structured k-SAT instances
SAT'08 Proceedings of the 11th international conference on Theory and applications of satisfiability testing
Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses
Proceedings of the forty-second ACM symposium on Theory of computing
Known algorithms on graphs of bounded treewidth are probably optimal
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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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−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.