Finding $(s,d)$-Hypernetworks in F-Hypergraphs is NP-Hard

TL;DR

This paper proves that finding $(s,d)$-hypernetworks in acyclic F-hypergraphs is NP-hard, revealing a surprising complexity boundary and fundamental asymmetry between F-hypergraphs and B-hypergraphs, unlike previous results for general hypergraphs.

Contribution

It establishes the NP-hardness of the $(s,d)$-hypernetwork problem in acyclic F-hypergraphs, contrasting prior results and explaining the asymmetry with B-hypergraphs.

Findings

NP-hardness of the problem in acyclic F-hypergraphs

Contrast with linear-time algorithms in acyclic B-hypergraphs

Explanation of the fundamental asymmetry between F- and B-hypergraphs

Abstract

We consider the problem of computing an $(s,d)$-hypernetwork in an acyclic F-hypergraph. This is a fundamental computational problem arising in directed hypergraphs, and is a foundational step in tackling problems of reachability and redundancy. This problem was previously explored in the context of general directed hypergraphs (containing cycles), where it is NP-hard, and acyclic B-hypergraphs, where a linear time algorithm can be achieved. In a surprising contrast, we find that for acyclic F-hypergraphs the problem is NP-hard, which also implies the problem is hard in BF-hypergraphs. This is a striking complexity boundary given that F-hypergraphs and B-hypergraphs would at first seem to be symmetrical to one another. We provide the proof of complexity and explain why there is a fundamental asymmetry between the two classes of directed hypergraphs.