FPT Approximation of Generalised Hypertree Width for Bounded Intersection Hypergraphs (2309.17049v2)

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 $ghw$.