FPT Parameterisations of Fractional and Generalised Hypertree Width

Matthias Lanzinger; Igor Razgon; Daniel Unterberger

Home/Archive/arXiv

arXiv · 2507.11080

FPT Parameterisations of Fractional and Generalised Hypertree Width

Matthias Lanzinger·Igor Razgon·Daniel Unterberger

View original source ↗Read source document ↗Export metadata ↓

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

Explore related subjects

cs.DS cs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Matthias Lanzinger, Igor Razgon, Daniel Unterberger. 2025-07-15. FPT Parameterisations of Fractional and Generalised Hypertree Width. https://arxiv.org/abs/2507.11080

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗

arXiv2026

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

cs.DS↗

arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗