Myhill-Nerode methods for hypergraphs

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.

💡 Research Summary

The paper extends the classic Myhill‑Nerode theorem, originally formulated for formal languages and later adapted to graphs, to the domain of hypergraphs. By viewing a hypergraph through its incidence graph—a bipartite representation linking vertices and hyperedges—the authors define a notion of “boundary states” that capture the interaction of a hypergraph fragment with the rest of the structure. They prove that, when the incidence treewidth (the treewidth of this bipartite graph) is bounded, only finitely many such boundary states exist up to an appropriate equivalence relation. This finiteness is the hypergraph analogue of the Myhill‑Nerode congruence and provides a systematic way to design dynamic‑programming algorithms that run in time linear in the size of the hypergraph, with a multiplicative factor depending only on the treewidth parameter.

Using this framework, the authors address two central hypergraph problems that are NP‑hard in general.

1. Hypergraph Cutwidth – The cutwidth of a hypergraph is the minimum, over all linear layouts of its vertices, of the maximum number of hyperedges that cross any cut between consecutive vertices. For graphs, cutwidth is known to be fixed‑parameter tractable (FPT) when parameterized by treewidth. The paper shows that, for hypergraphs, if the incidence treewidth is a constant k, then the cutwidth can be decided in O(f(k)·n) time, where n is the number of vertices and f(k) is a function independent of n. The algorithm proceeds by constructing a tree decomposition of the incidence graph, enumerating all possible configurations of the “port” vertices at each bag, and propagating minimal cutwidth values via a bottom‑up dynamic program. Because the number of configurations is bounded solely by k, the overall running time is linear in the input size, establishing that hypergraph cutwidth is fixed‑parameter linear with respect to the cutwidth bound k.

2. (Fractional/Generalized) Hypertree Width – Hypertree width measures how well a hypergraph can be decomposed into a tree‑like structure where each bag is covered by a small number of hyperedges; fractional and generalized variants relax the covering condition. The authors prove that the property “the hypertree width (or its fractional/generalized version) is at most t” cannot be expressed in monadic second‑order logic (MSO) even when the incidence treewidth is bounded. Their proof constructs an infinite family of hypergraphs with bounded incidence treewidth but pairwise distinct hypertree‑width values, and shows that any MSO formula would have to distinguish infinitely many non‑equivalent structures, contradicting the finite‑model property of MSO on bounded‑treewidth graphs. Consequently, these hypertree‑width problems are not MSO‑definable, which strongly suggests they are **W