Hardness of Parameterized Resolution
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.
Proof complexity
parameterized complexity
Resolution
Prover-Delayer Games
1-28
Regular Paper
Olaf
Beyersdorff
Olaf Beyersdorff
Nicola
Galesi
Nicola Galesi
Massimo
Lauria
Massimo Lauria
10.4230/DagSemProc.10061.4
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