Satisfiable Tseitin Formulas Are Hard for Nondeterministic Read-Once Branching Programs

We show that assuming the Exponential Time Hypothesis, the Partial Minimum Branching Program Size Problem (MBPSP * ) requires superpolynomial time. This result also applies to the partial minimization problems for many interesting subclasses of branching programs, such as read-k branching programs and OBDDs.Combining these results with our recent result (Glinskih and Riazanov, LATIN 2022) we obtain an unconditional superpolynomial lower bound on the size of Read-Once Nondeterministic Branching Programs (1-NBP) computing the total versions of the minimum BP, read-k-BP, and OBDD size problems.Additionally we show that it is NP-hard to check whether a given BP computing a partial Boolean function can be compressed to a BP of a given size.1 The same hardness result holds for Partial MFSP (MFSP * ).2 Suppose that f is such a reduction from MCSP s=3n (the language of the truth-tables computable by circuits with at most 3n gates) to MBPSP s=n 2.1 . Then taking any function requiring circuits of size larger than 3n [FGHK16, LY22] and applying f to it yields n 2.1 lower bound for general branching programs superior to the current state of the art. In a similar way, we get superlinear circuit lower bounds from any reduction from MBPSP s=n 2 to MCSP s=n 1+ε .3 We note that results in this area often establish NP-hardness and even hardness of size approximation for the case of concise representation, while our result only establishes ETH-hardness of the exact version of the problem. 4 The input to this problem has Θ(n 4 log n) bit size.

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Characterizing Tseitin-Formulas with Short Regular Resolution Refutations

Colnet

Mengel

2023

jair

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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)|). It follows that unsatisfiable Tseitin-formulas with polynomial length of regular resolution refutations are completely determined by the treewidth of the underlying graphs when these graphs have bounded degree. To prove this, we show that any regular resolution refutation of an unsatisfiable Tseitin-formula with graph G of bounded degree has length 2Ω(tw(G))/|V (G)|, thus essentially matching the known 2O(tw(G))poly(|V (G)|) upper bound. 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 2Ω(tw(G)) which yields our lower bound for regular resolution.

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Satisfiable Tseitin Formulas Are Hard for Nondeterministic Read-Once Branching Programs

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Characterizing Tseitin-Formulas with Short Regular Resolution Refutations

Characterizing Tseitin-Formulas with Short Regular Resolution Refutations

MCSP is Hard for Read-Once Nondeterministic Branching Programs

Characterizing Tseitin-Formulas with Short Regular Resolution Refutations

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