Parameterized Complexity of DPLL Search

We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires n steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [16] of understanding the Resolution complexity of this family of formulas. ∗Research was supported by the grant “Limits of Theorem Proving” from the John Templeton Foundation and a DAAD grant. Part of this work was done while the first author was visiting Sapienza University of Rome. A preliminary version of the results in this paper appeared in the proceedings of SAT’11 [14]. †Partly supported by Sapienza Research Project: Complessità e Rappresentabilità Compatta di Strutture Discrete. ‡Partially supported by the Eduard Čech Center for Algebra and Geometry (http://ecc.sci.muni.cz/), Czech Republic, and also funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC grant agreement no. 279611.