BOOK OF ABSTRACTS

Zusammenfassung: We study the performance of DPLL algorithms on parameterized pro-blems. 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 proce-dures. 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 nΩ(k) steps for a non-trivial distribution of graphs close to the critical threshold. These results are presented in the general framework for parameterized proof complexity as introduced by Dantchev, Martin, and Szeider [3]. 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. In our paper [2] we significantly improve 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