The Complexity of Weak Partition Connectivity in Hedgegraphs

We prove that the integer-threshold decision problem for weak partition connectivity in hedgegraphs is NP-complete, answering an open question about its computational complexity. Hardness holds even for connected unweighted hedgegraphs in which every hedge consists of exactly two nonempty, vertex-disjoint hyperedges whose union is the entire vertex set. On the same class of instances, hedge connectivity has a simple exact formula. Using a binary matrix representation, we express fractional weak partition connectivity as $m-ρ(A)$, where $ρ(A)$ maximizes the ratio of the number of selected rows to one less than the number of distinct projected columns. This formula yields both the hardness reduction and deterministic algorithms: exact computation when some reference column gives row supports satisfying a linear intersection condition, including the case of minimum row-support number $s(A)\le2$, and a partition-output polynomial-time approximation scheme (PTAS) for both the integer and fractional objectives on all full-support split systems. Unless $\mathrm{P}=\mathrm{NP}$, neither objective admits a fully polynomial-time approximation scheme (FPTAS) on this class.