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Lower Bounds on the Complexity of MSO_1 Model-Checking

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



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Robert Ganian
Petr Hlineny
Alexander Langer
Jan Obdržálek
Peter Rossmanith
Somnath Sikdar

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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)
https://doi.org/10.4230/LIPIcs.STACS.2012.326

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.
Keywords
  • Monadic Second-Order Logic
  • Treewidth
  • Lower Bounds
  • Exponential Time Hypothesis
  • Parameterized Complexity

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