Exponential lower bounds for the pigeonhole principle
Computational Complexity
An exponential lower bound to the size of bounded depth Frege proofs of the Pigeonhole Principle
Random Structures & Algorithms
The Efficiency of Resolution and Davis--Putnam Procedures
SIAM Journal on Computing
Describing parameterized complexity classes
Information and Computation
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Proof Complexity
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
The Resolution Complexity of Independent Sets and Vertex Covers in Random Graphs
Computational Complexity
Data reductions, fixed parameter tractability, and random weighted d-CNF satisfiability
Artificial Intelligence
Resolution Is Not Automatizable Unless W[P] Is Tractable
SIAM Journal on Computing
Parameterized complexity of DPLL search procedures
SAT'11 Proceedings of the 14th international conference on Theory and application of satisfiability testing
Parameterized Complexity
Parameterized Bounded-Depth Frege Is not Optimal
ACM Transactions on Computation Theory (TOCT)
Some definitorial suggestions for parameterized proof complexity
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
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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.