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 $ghw$.

翻译：广义超树宽度（$ghw$）是一个超图参数，对于许多具有自然超图结构的重要问题的可解性至关重要。计算超图的$ghw$具有公认的难度。该问题的判定版本（检查是否$ghw(H) \leq k$）以$k$为参数时是paraNP难的。此外，$ghw$的近似难度至少与Set-Cover的近似难度相当，而后者已知不允许任何FPT近似算法。当前对$ghw$计算的研究主要集中于识别超图的限制性结构（例如边交大小上界）使得$ghw$存在XP算法。然而，即使在这些限制下，该问题至今仍无任何类型的FPT算法。本文首次向$ghw$的FPT算法迈出一步，证明了对于有界边交大小的图，该参数可在FPT时间内近似。具体而言，我们证明存在一个以$k$和$d$为参数的FPT算法：对于输入超图$H$（其边交最大基数为$d$）和整数$k$，该算法要么输出树分解且满足$ghw(H) \leq 4k(k+d+1)(2k-1)$，要么拒绝（此时保证$ghw(H) > k$）。因此，在有界边交超图这一特殊情况下，我们得到了$ghw$的FPT $O(k^3)$-近似算法。