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Parameterized BoundedDepth Frege is Not Optimal
, 2012
"... A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [11]. There the authors show important results on treelike Parameterized Resolution—a parameterized version of classical Resolution—and their gap complexity theorem implies lower bounds for that s ..."
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Cited by 7 (3 self)
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A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [11]. There the authors show important results on treelike 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 n Ω(k) in parameterized boundeddepth Frege, and, as a special case, in daglike Parameterized Resolution. This answers an open question posed in [11]. In the opposite direction, we interpret a wellknown 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 boundedwidth CNF’s.
Completely inapproximable monotone and antimonotone parameterized problems
"... We prove that weighted monotone/antimonotone circuit satisfiability has no fixedparameter tractable approximation algorithm with any approximation ratio function ρ, unless FPT 6 = W [1]. In particular, not having such an fptapproximation algorithm implies that these problems have no polynomialti ..."
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Cited by 6 (0 self)
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We prove that weighted monotone/antimonotone circuit satisfiability has no fixedparameter tractable approximation algorithm with any approximation ratio function ρ, unless FPT 6 = W [1]. In particular, not having such an fptapproximation algorithm implies that these problems have no polynomialtime approximation algorithms with ratio ρ(OPT) for any nontrivial function ρ.