Parameterized Bounded-Depth Frege Is Not Optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. There the authors concentrate on tree-like Parameterized Resolution--a parameterized version of classical Resolution--and their gap complexity theorem implies lower bounds for that system. The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of boundeddepth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size nΩ(k) in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that treelike Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's.

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