Efficient Approximation of Fractional Hypertree Width

TL;DR

This paper introduces two polynomial-time approximation algorithms for fractional hypertree width, achieving better bounds and efficiency than previous methods, with implications for hypergraph decomposition and related computational problems.

Contribution

The paper presents the first non-trivial polynomial-time approximation algorithms for fractional hypertree width and generalized hypertree width, improving both approximation ratios and running times.

Findings

First polynomial-time approximation for fractional hypertree width.

Improved approximation ratio of O(ω log n log ω).

Enhanced algorithms for hypergraphs with bounded intersection property.

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 $\omega$, runs in polynomial time and produces a tree decomposition of $H$ of fractional hypertree width $O(\omega \log n \log \omega)$. As an immediate corollary this yields polynomial time $O(\log^2 n \log \omega)$-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 $\omega$ 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 $n^{\omega}m^{O(1)}$ and produces a tree decomposition of $H$ of fractional hypertree width $O(\omega \log^2 \omega)$. This significantly improves over the $(n+m)^{O(\omega^3)}$ time algorithm of Marx [ACM TALG 2010], which produces a tree decomposition of fractional hypertree width $O(\omega^3)$, 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.