发表机构
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明hedgegraph中弱划分连通性的整数阈值判定问题是NP完全的，并给出分数目标的精确公式及PTAS，同时证明除非P=NP否则无FPTAS。
AI 中文摘要
我们证明了在hedgegraph中，弱划分连通性的整数阈值判定问题是NP完全的，回答了关于其计算复杂性的一个开放问题。即使对于连通的未加权hedgegraph，其中每个hedge恰好由两个非空、顶点不相交的超边组成，且这两个超边的并集是整个顶点集，困难性依然成立。在同一类实例上，hedge连通性有一个简单的精确公式。利用二元矩阵表示，我们将分数弱划分连通性表示为$m-\rho(A)$，其中$\rho(A)$最大化所选行数与不同投影列数减一之比。该公式同时给出了困难性归约和确定性算法：当某个参考列使得行支撑满足线性交集条件时（包括最小行支撑数$s(A)\le2$的情形），可精确计算；并且对所有全支撑分裂系统，给出了一个输出划分的多项式时间近似方案（PTAS），适用于整数和分数目标。除非$\mathrm{P}=\mathrm{NP}$，否则在该类上，这两个目标均不存在完全多项式时间近似方案（FPTAS）。
英文摘要
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.
