Abstract
This paper settles the computational complexity of model checking of several extensions of the monadic second order (MSO) logic on two classes of graphs: graphs of bounded treewidth and graphs of bounded neighborhood diversity.
A classical theorem of Courcelle states that any graph property definable in MSO is decidable in linear time on graphs of bounded treewidth. Algorithmic metatheorems like Courcelle’s serve to generalize known positive results on various graph classes. We explore and extend three previously studied MSO extensions: global and local cardinality constraints (CardMSO and MSO-LCC) and optimizing a fair objective function (fairMSO).
We show how these fragments relate to each other in expressive power and highlight their (non)linearity. On the side of neighborhood diversity, we show that combining the linear variants of local and global cardinality constraints is possible while keeping FPT runtime but removing linearity of either makes this impossible, and we provide an XP algorithm for the hard case. Furthemore, we show that even the combination of the two most powerful fragments is solvable in polynomial time on graphs of bounded treewidth.
This research was partially supported by project 338216 of GA UK and the grant SVV–2017–260452. M. Koutecký was also supported by the project GA15-11559S of GA ČR. D. Knop and T. Masařík were also supported by the project GA17-09142S of GA ČR.
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Knop, D., Koutecký, M., Masařík, T., Toufar, T. (2017). Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_26
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