Parameterized Proof Complexity: a Complexity Gap for Parameterized Tree-like Resolution

We propose a proof-theoretic approach for gaining evidence that certain parameterized problems are not fixed-parameter tractable. We consider proofs that witness that a given propositional formula cannot be satisfied by a truth assignment that sets at most k variables to true, considering k as the parameter. One could separate the parameterized complexity classes FPT and W[2] by showing that there is no proof system (for CNF formulas) that admits proofs of size f(k)n where f is a computable function and n represents the size of the propositional formula. We provide a first step and show that tree-like resolution does not admit such proofs. We obtain this result as a corollary to a meta-theorem, the main result of this paper. The meta-theorem extends Riis’ Complexity Gap Theorem for tree-like resolution. Riis’ result establishes a dichotomy between polynomial and exponential size tree-like resolution proofs for propositional formulas that uniformly encode a first-order principle over a universe of size n: (1) either there are tree-like resolution proofs of size polynomial in n, or (2) the proofs have size at least 2 for some constant ε; the second case prevails exactly when the first-order principle has no finite but some infinite model. We show that the parameterized setting allows a refined classification, splitting the second case into two subcases: (2a) there are tree-like resolution proofs of size at most βn for some constants α, β; or (2b) every tree-like resolution proof has size at least n γ for some constant 0 < γ ≤ 1; the latter case prevails exactly if for every infinite model, a certain associated hypergraph has no finite dominating set. We provide examples of first-order principles for all three cases. Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 1 (2007)