FPT Approximation of Generalised Hypertree Width for Bounded Intersection Hypergraphs
Symposium on Theoretical Aspects of Computer Science(2023)
摘要
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) ≤ 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) ≤ 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.
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