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DOI: 10.4230/LIPIcs.STACS.2012.326
Lower Bounds on the Complexity of MSO_1 Model-Checking

Authors: Robert Ganian, Petr Hlineny, Alexander Langer, Jan Obdržálek, Peter Rossmanith, and Somnath Sikdar

Published in: LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

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.
Cite as

Robert Ganian, Petr Hlineny, Alexander Langer, Jan Obdržálek, Peter Rossmanith, and Somnath Sikdar. Lower Bounds on the Complexity of MSO_1 Model-Checking. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 326-337, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{ganian_et_al:LIPIcs.STACS.2012.326,
  author =	{Ganian, Robert and Hlineny, Petr and Langer, Alexander and Obdr\v{z}\'{a}lek, Jan and Rossmanith, Peter and Sikdar, Somnath},
  title =	{{Lower Bounds on the Complexity of MSO\underline1 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 =	{D\"{u}rr, Christoph and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.326},
  URN =		{urn:nbn:de:0030-drops-34185},
  doi =		{10.4230/LIPIcs.STACS.2012.326},
  annote =	{Keywords: Monadic Second-Order Logic, Treewidth, Lower Bounds, Exponential Time Hypothesis, Parameterized Complexity}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2009.2326
A Fine-grained Analysis of a Simple Independent Set Algorithm

Authors: Joachim Kneis, Alexander Langer, and Peter Rossmanith

Published in: LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Abstract
We present a simple exact algorithm for the \is\ problem with a runtime bounded by $O(\rt^n \poly(n))$. This bound is obtained by, firstly, applying a new branching rule and, secondly, by a distinct, computer-aided case analysis. The new branching rule uses the concept of satellites and has previously only been used in an algorithm for sparse graphs. The computer-aided case analysis allows us to capture the behavior of our algorithm in more detail than in a traditional analysis. The main purpose of this paper is to demonstrate how a very simple algorithm can outperform more complicated ones if the right analysis of its running time is performed.
Cite as

Joachim Kneis, Alexander Langer, and Peter Rossmanith. A Fine-grained Analysis of a Simple Independent Set Algorithm. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 287-298, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{kneis_et_al:LIPIcs.FSTTCS.2009.2326,
  author =	{Kneis, Joachim and Langer, Alexander and Rossmanith, Peter},
  title =	{{A Fine-grained Analysis of a Simple Independent Set Algorithm}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{287--298},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-13-2},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{4},
  editor =	{Kannan, Ravi and Narayan Kumar, K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2326},
  URN =		{urn:nbn:de:0030-drops-23269},
  doi =		{10.4230/LIPIcs.FSTTCS.2009.2326},
  annote =	{Keywords: Exact Algorithms, Independent Set, Computer-aided Proof}
}
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