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Lower Bounds on the Complexity of MSO_1 Model-Checking

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Abstract

One of the most important algorithmic meta-theorems is a famous result by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (MSO2) is decidable in linear time on any class of graphs of bounded tree-width. In the parlance of parameterized complexity, this means that MSO2 model-checking is fixed-parameter tractable with respect to the tree-width as parameter. Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound---that MSO2 model-checking is not even in XP wrt the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH). In this paper we present a closely related result. We show that even MSO1 model-checking with a fixed set of vertex labels, but without edge-set quantifications, is not in XP wrt the formula size as parameter for graph classes which are subgraph-closed and whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari, (1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2) we assume a different set of problems to be efficiently decidable, namely MSO1-definable properties on vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs. Our result has an interesting consequence in the realm of digraph width measures: Strengthening a recent result, we show that no subdigraph-monotone measure can be algorithmically useful, unless it is within a poly-logarithmic factor of (undirected) tree-width.

BibTeX - Entry

@InProceedings{ganian_et_al:LIPIcs:2012:3418,
  author =	{Robert Ganian and Petr Hlineny and Alexander Langer and Jan Obdrž{\'a}lek and Peter Rossmanith and Somnath Sikdar},
  title =	{{Lower Bounds on the Complexity of MSO_1 Model-Checking}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{326--337},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{Christoph D{\"u}rr and Thomas Wilke},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2012/3418},
  URN =		{urn:nbn:de:0030-drops-34185},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2012.326},
  annote =	{Keywords: Monadic Second-Order Logic, Treewidth, Lower Bounds, Exponential Time Hypothesis, Parameterized Complexity}
}

Keywords: Monadic Second-Order Logic, Treewidth, Lower Bounds, Exponential Time Hypothesis, Parameterized Complexity
Seminar: 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)
Issue date: 2012
Date of publication: 2012


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