Characterizing Tseitin-formulas with short regular resolution refutations

Tseitin-formulas are systems of parity constraints whose structure is described by a graph. These formulas have been studied extensively in proof complexity as hard instances in many proof systems. In this paper, we prove that a class of unsatisfiable Tseitin-formulas of bounded degree has regular resolution refutations of polynomial length if and only if the treewidth of all underlying graphs G for that class is in O(log |V (G)|). To do so, we show that any regular resolution refutation of an unsatisfiable Tseitin-formula with graph G of bounded degree has length 2/|V (G)|, thus essentially matching the known 2poly(|V (G)|) upper bound up. Our proof first connects the length of regular resolution refutations of unsatisfiable Tseitin-formulas to the size of representations of satisfiable Tseitin-formulas in decomposable negation normal form (DNNF). Then we prove that for every graph G of bounded degree, every DNNF-representation of every satisfiable Tseitin-formula with graph G must have size 2 which yields our lower bound for regular resolution.

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