Myhill-Nerode methods for hypergraphs
Abstract
We give an analog of the Myhill-Nerode methods from formal language theory for hypergraphs and use it to derive the following results for two NP-hard hypergraph problems: * We provide an algorithm for testing whether a hypergraph has cutwidth at most k that runs in linear time for constant k. In terms of parameterized complexity theory, the problem is fixed-parameter linear parameterized by k. * We show that it is not expressible in monadic second-order logic whether a hypergraph has bounded (fractional, generalized) hypertree width. The proof leads us to conjecture that, in terms of parameterized complexity theory, these problems are W[1]-hard parameterized by the incidence treewidth (the treewidth of the incidence graph). Thus, in the form of the Myhill-Nerode theorem for hypergraphs, we obtain a method to derive linear-time algorithms and to obtain indicators for intractability for hypergraph problems parameterized by incidence treewidth. In an appendix, we point out an error and a fix to the proof of the Myhill-Nerode theorem for graphs in Downey and Fellow's book on parameterized complexity.
Cite
@article{arxiv.1211.1299,
title = {Myhill-Nerode methods for hypergraphs},
author = {René van Bevern and Rodney G. Downey and Michael R. Fellows and Serge Gaspers and Frances A. Rosamond},
journal= {arXiv preprint arXiv:1211.1299},
year = {2016}
}
Comments
A preliminary version of this article appeared in the proceedings of ISAAC 2013. This extended and revised version contains the full proof details, more figures, and corollaries to make the application of the Myhill-Nerode theorem for hypergraphs easier in an algorithmic setting. Moreover, it provides a fix to the proof of the Myhill-Nerode theorem for graphs in the books of Downey and Fellows