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Lower Bounds on the Complexity of MSO_1 Model-Checking Ganian, Robert Hlineny, Petr Langer, Alexander Obdržálek, Jan Rossmanith, Peter Sikdar, Somnath Monadic Second-Order Logic Treewidth Lower Bounds Exponential Time Hypothesis Parameterized Complexity 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Robert Ganian and Petr Hlineny and Alexander Langer and Jan Obdržálek and Peter Rossmanith and Somnath Sikdar 2012 Is Part Of LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2012.326 urn:nbn:de:0030-drops-34185 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.326 eng https://creativecommons.org/licenses/by-nd/3.0/legalcode