FPT Parameterisations of Fractional and Generalised Hypertree Width

TL;DR

This paper introduces the first fixed-parameter tractable algorithms for exactly computing generalized and fractional hypertree widths in hypergraphs, broadening the scope of efficient algorithms for hypergraph width measures.

Contribution

It develops FPT algorithms for generalized and fractional hypertree width, extending previous work on treewidth and introducing new combinatorial tools for hypergraph width measures.

Findings

First exact FPT algorithms for ghw and fhw

Applicable to a broad class of width functions

Framework extends to discretized adaptive width

Abstract

We present the first fixed-parameter tractable (FPT) algorithms for exact computation of generalized hypertree width (ghw) and fractional hypertree width (fhw). Our algorithms are parameterized by the target width, the rank, and the maximum degree of the input hypergraph. More generally, we show that testing f-width is in FPT for a broad class of width functions that we call manageable. This class contains the edge cover number $\rho$ and its fractional relaxation $\rho^*$, and thus covers both generalized and fractional hypertree width. We additionally extend our framework to also obtain an fpt algorithm for computing a discretized version of adaptive width. Our approach extends a recent algorithm for treewidth (Boja\'ncyk & Pilipczuk, LMCS 2022) that utilises monadic second-order transductions. To extend this idea beyond treewidth we develop new combinatorial machinery around elimination forests in hypergraphs, culminating in a structural normal form for optimal witnesses that makes transduction-based optimisation applicable in the much more general context of manageable width functions. This yields the first exact FPT algorithms for these measures under any nontrivial parameterisation and provides structural tools that may enable more direct optimisation algorithms