On the complexity of strongly connected components in directed hypergraphs

TL;DR

This paper introduces an almost linear time algorithm for computing terminal strongly connected components in directed hypergraphs, highlighting the increased complexity of SCCs compared to standard graphs and their relevance to computational geometry.

Contribution

It presents a nearly linear time algorithm for terminal SCCs in directed hypergraphs and establishes complexity bounds showing SCC computation is harder than in directed graphs.

Findings

Almost linear time algorithm for terminal SCCs

Superlinear lower bound on reachability transitive reduction

Linear time reduction from minimal set problem

Abstract

We study the complexity of some algorithmic problems on directed hypergraphs and their strongly connected components (SCCs). The main contribution is an almost linear time algorithm computing the terminal strongly connected components (i.e. SCCs which do not reach any components but themselves). "Almost linear" here means that the complexity of the algorithm is linear in the size of the hypergraph up to a factor alpha(n), where alpha is the inverse of Ackermann function, and n is the number of vertices. Our motivation to study this problem arises from a recent application of directed hypergraphs to computational tropical geometry. We also discuss the problem of computing all SCCs. We establish a superlinear lower bound on the size of the transitive reduction of the reachability relation in directed hypergraphs, showing that it is combinatorially more complex than in directed graphs. Besides, we prove a linear time reduction from the well-studied problem of finding all minimal sets among a given family to the problem of computing the SCCs. Only subquadratic time algorithms are known for the former problem. These results strongly suggest that the problem of computing the SCCs is harder in directed hypergraphs than in directed graphs.