Descriptive Complexity, Canonisation, and Definable Graph Structure Theory

By Martin Grohe. 2017

209 citations recorded by the source corpus.

Descriptive complexity theory establishes a connection between the computational complexity of algorithmic problems (the computational resources required to solve the problems) and their descriptive complexity (the language resources required to describe the problems). This ground-breaking book approaches descriptive complexity from the angle of modern structural graph theory, specifically graph minor theory. It develops a ‘definable structure theory’ concerned with the logical definability of graph-theoretic concepts such as tree decompositions and embeddings. The first part starts with an introduction to the background, from logic, complexity, and graph theory, and develops the theory up to first applications in descriptive complexity theory and graph isomorphism testing. It may serve as the basis for a graduate-level course. The second part is more advanced and mainly devoted to the proof of a single, previously unpublished theorem: properties of graphs with excluded minors are decidable in polynomial time if, and only if, they are definable in fixed-point logic with counting.

Subfields: Computational Theory and Mathematics

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Citations in the corpus

Provably Powerful Graph Networks (2019, methodology): “The Weisfeiler Lehman tests is a hierarchy of increasingly powerful graph isomorphism tests (Grohe, 2017).”
Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks (2018, methodology): “…the Weisfeiler-Lehman subtree kernel (Shervashidze et al. 2011), which is based on the 1 - dimensional Weisfeiler-Leman graph isomorphism heuristic (Grohe 2017, pp. 79 ff.), generates node features through an iterative relabeling, or coloring , scheme: First, all nodes are assigned a common…”
Weisfeiler and Leman go sparse: Towards scalable higher-order graph embeddings (2019, methodology): “…this was reﬁned in (Maron et al., 2019a) by introducing k -order invariant graph networks ( k - IGN ), based on Maron et al. (Maron et al., 2019b), and references therein, which are equivalent to the folklore variant of the k - WL (Grohe, 2017) in terms of distinguishing non-isomorphic graphs.”