Myhill-Nerode methods for hypergraphs

TL;DR

This paper extends Myhill-Nerode methods to hypergraphs, providing linear-time algorithms for certain problems and showing inexpressibility results, thereby advancing the understanding of hypergraph problem complexity.

Contribution

It introduces a Myhill-Nerode framework for hypergraphs, enabling linear-time algorithms and complexity intractability indicators for hypergraph problems.

Findings

Linear-time algorithm for hypergraph cutwidth testing.

Inexpressibility of bounded hypertree width in monadic second-order logic.

Conjecture that certain hypergraph problems are W[1]-hard when parameterized by incidence treewidth.

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.