License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.MFCS.2019.49
URN: urn:nbn:de:0030-drops-109932
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Galesi, Nicola ; Itsykson, Dmitry ; Riazanov, Artur ; Sofronova, Anastasia

Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs

LIPIcs-MFCS-2019-49.pdf (0.5 MB)


We prove that there is a constant K such that Tseitin formulas for an undirected graph G requires proofs of size 2^{tw(G)^{Omega(1/d)}} in depth-d Frege systems for d<(K log n)/(log log n), where tw(G) is the treewidth of G. This extends Håstad recent lower bound for the grid graph to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2^{tw(G)^{O(1/d)}} poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution.

BibTeX - Entry

  author =	{Nicola Galesi and Dmitry Itsykson and Artur Riazanov and Anastasia Sofronova},
  title =	{{Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{49:1--49:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-109932},
  doi =		{10.4230/LIPIcs.MFCS.2019.49},
  annote =	{Keywords: Tseitin formula, treewidth, AC0-Frege}

Keywords: Tseitin formula, treewidth, AC0-Frege
Collection: 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)
Issue Date: 2019
Date of publication: 20.08.2019

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