License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/DagSemProc.10061.4
URN: urn:nbn:de:0030-drops-25254
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Beyersdorff, Olaf ; Galesi, Nicola ; Lauria, Massimo

Hardness of Parameterized Resolution

10061.GalesiNicola.Paper.2525.pdf (0.5 MB)


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.

BibTeX - Entry

  author =	{Beyersdorff, Olaf and Galesi, Nicola and Lauria, Massimo},
  title =	{{Hardness of Parameterized Resolution}},
  booktitle =	{Circuits, Logic, and Games},
  pages =	{1--28},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{10061},
  editor =	{Benjamin Rossman and Thomas Schwentick and Denis Th\'{e}rien and Heribert Vollmer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-25254},
  doi =		{10.4230/DagSemProc.10061.4},
  annote =	{Keywords: Proof complexity, parameterized complexity, Resolution, Prover-Delayer Games}

Keywords: Proof complexity, parameterized complexity, Resolution, Prover-Delayer Games
Collection: 10061 - Circuits, Logic, and Games
Issue Date: 2010
Date of publication: 26.04.2010

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