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FPT Parameterisations of Fractional and Generalised Hypertree Width

Matthias Lanzinger, Igor Razgon, Daniel Unterberger

TL;DR

The paper tackles the challenge of computing generalized and fractional hypertree widths (ghw and fhw) by introducing the class of manageable width functions and extending the Bojańczyk–Pilipczuk MSO-transduction framework to hypergraphs via elimination forests. It proves that f-width-check is fixed-parameter tractable for any manageable f, thereby yielding the first exact FPT algorithms for ghw and fhw under nontrivial parameterizations (k, rank(H), Δ(H)). A key structural result, the Reduced Hypergraph Dealternation Lemma, bounds how optimal witnesses interact with a backbone decomposition, enabling an MSO-based optimization pipeline. The framework further extends to discretized adaptive width and provides a factor-2 FPT approximation for a relaxed adaptive width, illustrating the broad applicability of the approach to a family of width measures. Overall, the work lays a solid theoretical foundation for exact and approximate FPT algorithms for hypergraph width measures and suggests avenues for practical algorithm design.

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 $ρ$ and its fractional relaxation $ρ^*$, 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 (Bojańcyk & 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

FPT Parameterisations of Fractional and Generalised Hypertree Width

TL;DR

The paper tackles the challenge of computing generalized and fractional hypertree widths (ghw and fhw) by introducing the class of manageable width functions and extending the Bojańczyk–Pilipczuk MSO-transduction framework to hypergraphs via elimination forests. It proves that f-width-check is fixed-parameter tractable for any manageable f, thereby yielding the first exact FPT algorithms for ghw and fhw under nontrivial parameterizations (k, rank(H), Δ(H)). A key structural result, the Reduced Hypergraph Dealternation Lemma, bounds how optimal witnesses interact with a backbone decomposition, enabling an MSO-based optimization pipeline. The framework further extends to discretized adaptive width and provides a factor-2 FPT approximation for a relaxed adaptive width, illustrating the broad applicability of the approach to a family of width measures. Overall, the work lays a solid theoretical foundation for exact and approximate FPT algorithms for hypergraph width measures and suggests avenues for practical algorithm design.

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 and its fractional relaxation , 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 (Bojańcyk & 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

Paper Structure

This paper contains 17 sections, 44 theorems, 12 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Technical Overview
  4. The Main Theorem
  5. The Structure of Optimal Elimination Forests
  6. Efficient Construction of Transductions
  7. Proof of Theorem \ref{['th:mainalgo']}
  8. Computation and Approximation of Adaptive Width Variants
  9. Conclusion & Future Work
  10. Proof of the Reduced Hypergraph Dealteration Lemma (Lemma \ref{['lem:dealt']})
  11. Bounding the Number of $X$-Intervals (Lemma \ref{['lem:boundedxintervals']})
  12. Neutral Swaps on Spines with Many Factors (Lemma \ref{['lem:homogenousneutral']})
  13. Gathering the things together
  14. Verifying Hypergraph Widths in MSO
  15. Proof of Theorem \ref{['th:cgchrom']}
  16. ...and 2 more sections

Key Result

Theorem 1

$f$-width-check is fixed parameter tractable for any manageable width function $f$.

Figures (4)

  • Figure 1: An elimination forest for the depicted hypergraph (left) and its induced tree decomposition (right).
  • Figure 2: Illustrations of factors and key tree decomposition notions used in this paper.
  • Figure 3: A swap operation exchanging two consecutive monochromatic intervals.
  • Figure 4: A swap $F'$ of $F$.

Theorems & Definitions (137)

  • Theorem 1
  • Theorem 2
  • Definition 2: Factors
  • Definition 2: Elimination Forest
  • Definition 3: Monotone separability
  • Definition 3
  • Theorem 4: Extended version of \ref{['thm1']}
  • Corollary 5
  • Definition 6
  • Definition 7
  • ...and 127 more