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Hardness of Parameterized Resolution Beyersdorff, Olaf Galesi, Nicola Lauria, Massimo Proof complexity parameterized complexity Resolution Prover-Delayer Games Parameterized Resolution and, moreover, a general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider (FOCS'07). In that paper, Dantchev et al. show a complexity gap in tree-like Parameterized Resolution for propositional formulas arising from translations of first-order principles. We broadly investigate Parameterized Resolution obtaining the following main results: 1) We introduce a purely combinatorial approach to obtain lower bounds to the proof size in tree-like Parameterized Resolution. For this we devise a new asymmetric Prover-Delayer game which characterizes proofs in (parameterized) tree-like Resolution. By exhibiting good Delayer strategies we then show lower bounds for the pigeonhole principle as well as the order principle. 2) Interpreting a well-known FPT algorithm for vertex cover as a DPLL procedure for Parameterized Resolution, we devise a proof search algorithm for Parameterized Resolution and show that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's. 3) We answer a question posed by Dantchev, Martin, and Szeider showing that dag-like Parameterized Resolution is not fpt-bounded. We obtain this result by proving that the pigeonhole principle requires proofs of size $n^{Omega(k)}$ in dag-like Parameterized Resolution. For this lower bound we use a different Prover-Delayer game which was developed for Resolution by Pudlák. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Olaf Beyersdorff and Nicola Galesi and Massimo Lauria 2010 Is Part Of Dagstuhl Seminar Proceedings, Volume 10061, Circuits, Logic, and Games (2010) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.10061.4 urn:nbn:de:0030-drops-25254 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10061.4 eng https://creativecommons.org/licenses/by/4.0/legalcode