Efficient Approximation of Fractional Hypertree Width

TL;DR

This work addresses the problem of approximating the fractional hypertree width ($\operatorname{fhtw}$) and related width measures for hypergraphs. It introduces two main algorithms: a polynomial-time method that, when $\operatorname{fhtw}(H)\le\omega$, outputs a tree decomposition with $\operatorname{fhtw}(H)\le O(\omega \log \omega \log n)$, and a second algorithm with running time $n^{\omega}m^{O(1)}$ achieving $\operatorname{fhtw}(H)\le O(\omega \log^2 \omega)$, improving over prior exponential-time approaches. The core technique is a novel rounding framework for fractional $(A,B)$-separators that converts them into small integral separators with controlled fractional edge-cover costs, combined with ball-growing and a LP-based balanced-separator methodology. The paper also integrates these components within a Robertson–Seymour style decomposition framework to obtain polylogarithmic-approximation guarantees for $\operatorname{fhtw}$ and related widths, including refined bounds when the incidence structure has bounded degeneracy or small maximum edge intersections. Additionally, a clique-Menger-like theorem and related results extend the utility of the rounding scheme beyond hypergraph widths. Overall, the work advances tractable approximation strategies for hypergraph width parameters, with implications for CSPs and database query optimization where such decompositions underpin efficient algorithms.

Abstract

We give two new approximation algorithms to compute the fractional hypertree width of an input hypergraph. The first algorithm takes as input $n$-vertex $m$-edge hypergraph $H$ of fractional hypertree width at most $ω$, runs in polynomial time and produces a tree decomposition of $H$ of fractional hypertree width $O(ω\log n \log ω)$. As an immediate corollary this yields polynomial time $O(\log^2 n \log ω)$-approximation algorithms for (generalized) hypertree width as well. To the best of our knowledge our algorithm is the first non-trivial polynomial-time approximation algorithm for fractional hypertree width and (generalized) hypertree width, as opposed to algorithms that run in polynomial time only when $ω$ is considered a constant. For hypergraphs with the bounded intersection property we get better bounds, comparable with that recent algorithm of Lanzinger and Razgon [STACS 2024]. The second algorithm runs in time $n^ωm^{O(1)}$ and produces a tree decomposition of $H$ of fractional hypertree width $O(ω\log^2 ω)$. This significantly improves over the $(n+m)^{O(ω^3)}$ time algorithm of Marx [ACM TALG 2010], which produces a tree decomposition of fractional hypertree width $O(ω^3)$, both in terms of running time and the approximation ratio. Our main technical contribution, and the key insight behind both algorithms, is a variant of the classic Menger's Theorem for clique separators in graphs: For every graph $G$, vertex sets $A$ and $B$, family ${\cal F}$ of cliques in $G$, and positive rational $f$, either there exists a sub-family of $O(f \cdot \log^2 n)$ cliques in ${\cal F}$ whose union separates $A$ from $B$, or there exist $f \cdot \log |{\cal F}|$ paths from $A$ to $B$ such that no clique in ${\cal F}$ intersects more than $\log |{\cal F}|$ paths.

Figures (1)

• Figure 1: Interval graphs $\mathcal{I}_2$ (left) and $\mathcal{I}_3$ (right), where we depict each pair of intervals $I_{j,0}^k$ and $I_{j,1}^k$ by a single interval $I_{j}^k$ for simplicity. In a green layer, we depict intervals $I_{1}^k$, in a blue layer $I_{2}^k$, and in a purple layer $I_{3}^k$, for all values of $k$. In $\mathcal{I}_3$, we highlight the spanning tree rooted in $I_{1,0}^0$ by yellow. The root has children $I_{2,0}^0$ and $I_{2,1}^0$ (both depicted as $I_{2}^0$ in the blue layer). The leaves are the children of $I_{2,0}^0$ ($I_{3,0}^0$ and $I_{3,1}^0$) and the children of $I_{2,1}^0$ ($I_{3,0}^1$ and $I_{3,1}^1$), see the purple layer.

Theorems & Definitions (61)

• Theorem 1.0
• Theorem 1.0
• Lemma 1.1
• Theorem 1.1: Clique Menger's Theorem
• proof : Proof sketch of Lemma \ref{['lem:mainCliqueSepSimplified']}
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Lemma 3.6