When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2017.2
URN: urn:nbn:de:0030-drops-75410
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Lauria, Massimo ; Nordström, Jakob

Graph Colouring is Hard for Algorithms Based on Hilbert's Nullstellensatz and Gröbner Bases

LIPIcs-CCC-2017-2.pdf (0.6 MB)


We consider the graph k-colouring problem encoded as a set of polynomial equations in the standard way. We prove that there are bounded-degree graphs that do not have legal k-colourings but for which the polynomial calculus proof system defined in [Clegg et al. '96, Alekhnovich et al. '02] requires linear degree, and hence exponential size, to establish this fact. This implies a linear degree lower bound for any algorithms based on Gröbner bases solving graph k-colouring} using this encoding. The same bound applies also for the algorithm studied in a sequence of papers [De Loera et al. '08, '09, '11, '15] based on Hilbert's Nullstellensatz proofs for a slightly different encoding, thus resolving an open problem mentioned, e.g., in [De Loera et al. '09] and [Li et al. '16]. We obtain our results by combining the polynomial calculus degree lower bound for functional pigeonhole principle (FPHP) formulas over bounded-degree bipartite graphs in [Miksa and Nordström '15] with a reduction from FPHP to k-colouring derivable by polynomial calculus in constant degree.

BibTeX - Entry

  author =	{Massimo Lauria and Jakob Nordstr{\"o}m},
  title =	{{Graph Colouring is Hard for Algorithms Based on Hilbert's Nullstellensatz and Gr{\"o}bner Bases}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{2:1--2:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{Ryan O'Donnell},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-75410},
  doi =		{10.4230/LIPIcs.CCC.2017.2},
  annote =	{Keywords: proof complexity, Nullstellensatz, Gr{\"o}bner basis, polynomial calculus, cutting planes, colouring}

Keywords: proof complexity, Nullstellensatz, Gröbner basis, polynomial calculus, cutting planes, colouring
Seminar: 32nd Computational Complexity Conference (CCC 2017)
Issue Date: 2017
Date of publication: 21.07.2017

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