The Complexity of Weak Partition Connectivity in Hedgegraphs

2026-09-13

AI Summary

研究解决了hedgegraphs中弱分区连通性的整数阈值决策问题的计算复杂性，证明其为NP完全，并提出了基于二进制矩阵表示的解决方法。

Abstract

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.

Problem

• weak partition connectivity
• hedgegraphs
• NP-complete
• computational complexity

Innovation

• NP-complete
• weak partition connectivity
• hedgegraphs
• polynomial-time approximation scheme (PTAS)
• binary matrix representation