FPT Approximation of Generalised Hypertree Width for Bounded Intersection Hypergraphs

TL;DR

This paper introduces the first fixed-parameter tractable (fpt) algorithm to approximate the generalized hypertree width of hypergraphs with bounded edge intersection, significantly advancing the computational understanding of this complex parameter.

Contribution

The paper presents the first fpt approximation algorithm for generalized hypertree width in hypergraphs with bounded edge intersection, achieving an O(k^3) approximation ratio.

Findings

Provides an fpt algorithm for approximating ghw in bounded intersection hypergraphs.

Achieves an O(k^3) approximation ratio for ghw.

Guarantees correct rejection when ghw exceeds parameter k.

Abstract

Generalised hypertree width ($ghw$) is a hypergraph parameter that is central to the tractability of many prominent problems with natural hypergraph structure. Computing $ghw$ of a hypergraph is notoriously hard. The decision version of the problem, checking whether $ghw(H) \leq k$, is paraNP-hard when parameterised by $k$. Furthermore, approximation of $ghw$ is at least as hard as approximation of Set-Cover, which is known to not admit any fpt approximation algorithms. Research in the computation of ghw so far has focused on identifying structural restrictions to hypergraphs -- such as bounds on the size of edge intersections -- that permit XP algorithms for $ghw$. Yet, even under these restrictions that problem has so far evaded any kind of fpt algorithm. In this paper we make the first step towards fpt algorithms for $ghw$ by showing that the parameter can be approximated in fpt time for graphs of bounded edge intersection size. In concrete terms we show that there exists an fpt algorithm, parameterised by $k$ and $d$, that for input hypergraph $H$ with maximal cardinality of edge intersections $d$ and integer $k$ either outputs a tree decomposition with $ghw(H) \leq 4k(k+d+1+)(2k-1)$, or rejects, in which case it is guaranteed that $ghw(H) > k$. Thus, in the special case, of hypergraphs of bounded edge intersection, we obtain an fpt $O(k^3)$-approximation algorithm for $ghw$.