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
Resolution and the Weak Pigeonhole Principle
CSL '97 Selected Papers from the11th International Workshop on Computer Science Logic
Optimality of size-width tradeoffs for resolution
Computational Complexity
Describing parameterized complexity classes
Information and Computation
Resolution lower bounds for the weak pigeonhole principle
Journal of the ACM (JACM)
Resolution lower bounds for perfect matching principles
Journal of Computer and System Sciences - Special issue on computational complexity 2002
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
Approximation of Natural W[P]-Complete Minimisation Problems Is Hard
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on 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 bounded-depth Frege is not optimal
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Parameterized Complexity
Parameterized Complexity of DPLL Search Procedures
ACM Transactions on Computational Logic (TOCL)
The complexity of proving that a graph is ramsey
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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A general framework for parameterized proof complexity was introduced by Dantchev et al. [2007]. There, the authors show important results 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 this article significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth 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 Dantchev et al. [2007]. 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 tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNFs.