Abstract
This paper is motivated by seeking the exact complexity of resolution refutation of Tseitin formulas. We prove that the size of any regular resolution refutation of a Tseitin formula \( {\rm T}(G, c)\) based on a connected graph \({G} =(V, E)\) is at least \(2^{\Omega({\rm tw}(G)/ \log |V|)}\), where \({\rm tw}(G)\) denotes the treewidth of a graph G. For constant-degree graphs, there is a known upper bound \(2^{\mathcal{O}({\rm tw}(G))}{\rm poly}(|V|)\) (Alekhnovich & Razborov Comput. Compl. 2011; Galesi, Talebanfard & Torán ACM Trans. Comput. Theory 2020), so our lower bound is tight up to a logarithmic factor in the exponent.
Our proof consists of two steps. First, we show that any regular resolution refutation of an unsatisfiable Tseitin formula \({\rm T}(G, c) \) of size S can be converted to a read-once branching program computing a satisfiable Tseitin formula \({\rm T}(G,c')\) of size \(S^{{\mathcal{O}}({\rm log} |V|)}\) and this bound is tight.
Second, we give the exact characterization of the nondeterministic read-once branching program (1-NBP) complexity of satisfiable Tseitin formulas in terms of structural properties of underlying graphs. Namely, we introduce a new graph measure, the component width (compw) and show that the size of a minimal \({1\text{-}\mathrm{NBP}}\) computing a satisfiable Tseitin formula \({\rm T}(G,c')\) based on a graph \({G} = (V, E)\) equals \(2^{compw}(G)\) up to a polynomial factor. Then we show that \(\Omega({\rm tw}(G)) \le {\rm compw}(G) \le {\mathcal{O}}({\rm tw}(G){\rm log}(|V|))\) and both of these bounds are tight. The lower bound improves the recent result by Glinskih & Itsykson (Theory Comput. Syst. 2021).
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17 November 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00037-021-00216-z
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Acknowledgements
The authors thank Ludmila Glinskih, Anastasia Sofronova, Svyatoslav Gryaznov, Alexander Knop, Alexander Smal and Navid Talebanfard for fruitful discussions and the reviewers for their detailed and useful comments. Dmitry Itsykson was partially supported by Young Russian Mathematics award.
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Itsykson, D., Riazanov, A., Sagunov, D. et al. Near-Optimal Lower Bounds on Regular Resolution Refutations of Tseitin Formulas for All Constant-Degree Graphs. comput. complex. 30, 13 (2021). https://doi.org/10.1007/s00037-021-00213-2
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DOI: https://doi.org/10.1007/s00037-021-00213-2