License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2019.11
URN: urn:nbn:de:0030-drops-114721
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Dreier, Jan ; Rossmanith, Peter

Hardness of FO Model-Checking on Random Graphs

LIPIcs-IPEC-2019-11.pdf (0.5 MB)


It is known that FO model-checking is fixed-parameter tractable on Erdös - Rényi graphs G(n,p(n)) if the edge-probability p(n) is sufficiently small [Grohe, 2001] (p(n)=O(n^epsilon/n) for every epsilon>0). A natural question to ask is whether this result can be extended to bigger probabilities. We show that for Erdös - Rényi graphs with vertex colors the above stated upper bound by Grohe is the best possible.
More specifically, we show that there is no FO model-checking algorithm with average FPT run time on vertex-colored Erdös - Rényi graphs G(n,n^delta/n) (0 < delta < 1) unless AW[*]subseteq FPT/poly. This might be the first result where parameterized average-case intractability of a natural problem with a natural probability distribution is linked to worst-case complexity assumptions.
We further provide hardness results for FO model-checking on other random graph models, including G(n,1/2) and Chung-Lu graphs, where our intractability results tightly match known tractability results [E. D. Demaine et al., 2014]. We also provide lower bounds on the size of shallow clique minors in certain Erdös - Rényi and Chung - Lu graphs.

BibTeX - Entry

  author =	{Jan Dreier and Peter Rossmanith},
  title =	{{Hardness of FO Model-Checking on Random Graphs}},
  booktitle =	{14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-129-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{148},
  editor =	{Bart M. P. Jansen and Jan Arne Telle},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-114721},
  doi =		{10.4230/LIPIcs.IPEC.2019.11},
  annote =	{Keywords: random graphs, FO model-checking}

Keywords: random graphs, FO model-checking
Collection: 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)
Issue Date: 2019
Date of publication: 04.12.2019

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