The monadic second-order logic of graphs. I. recognizable sets of finite graphs
Information and Computation
Upper bounds to the clique width of graphs
Discrete Applied Mathematics
Deciding first-order properties of locally tree-decomposable structures
Journal of the ACM (JACM)
Approximation Schemes for First-Order Definable Optimisation Problems
LICS '06 Proceedings of the 21st Annual IEEE Symposium on Logic in Computer Science
Problems parameterized by treewidth tractable in single exponential time: a logical approach
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
Twin-Cover: beyond vertex cover in parameterized algorithmics
IPEC'11 Proceedings of the 6th international conference on Parameterized and Exact Computation
Algorithmic Meta-theorems for Restrictions of Treewidth
Algorithmica - Special Issue: Parameterized and Exact Computation, Part I
When trees grow low: shrubs and fast MSO1
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Width Parameters Beyond Tree-width and their Applications
The Computer Journal
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A well-known result by Frick and Grohe shows that deciding FO logic on trees involves a parameter dependence that is a tower of exponentials. Though this lower bound is tight for Courcelle's theorem, it has been evaded by a series of recent meta-theorems for other graph classes. Here we provide some additional non-elementary lower bound results, which are in some senses stronger. Our goal is to explain common traits in these recent meta-theorems and identify barriers to further progress. More specifically, first, we show that on the class of threshold graphs, and therefore also on any union and complement-closed class, there is no model-checking algorithm with elementary parameter dependence even for FO logic. Second, we show that there is no model-checking algorithm with elementary parameter dependence for MSO logic even restricted to paths (or equivalently to unary strings), unless EXP=NEXP. As a corollary, we resolve an open problem on the complexity of MSO model-checking on graphs of bounded max-leaf number. Finally, we look at MSO on the class of colored trees of depth d. We show that, assuming the ETH, for every fixed d≥1 at least d+1 levels of exponentiation are necessary for this problem, thus showing that the (d+1)-fold exponential algorithm recently given by Gajarský and Hliněnỳ is essentially optimal.