Efficient Algorithms for Measuring the Funnel-likeness of DAGs

TL;DR

This paper introduces algorithms to measure how close a DAG is to a funnel, a special subclass of DAGs, by computing the arc-deletion distance, and demonstrates their effectiveness on various graphs.

Contribution

The paper presents the first algorithms for efficiently computing the arc-deletion distance to a funnel in DAGs, including exact and approximation methods.

Findings

Algorithms successfully compute arc-deletion distance on synthetic graphs.

Approximation algorithms perform well on real-world graphs.

Funnel-like structures enable polynomial-time solutions for certain problems.

Abstract

Funnels are a new natural subclass of DAGs. Intuitively, a DAG is a funnel if every source-sink path can be uniquely identified by one of its arcs. Funnels are an analog to trees for directed graphs that is more restrictive than DAGs but more expressive than in-/out-trees. Computational problems such as finding vertex-disjoint paths or tracking the origin of memes remain NP-hard on DAGs while on funnels they become solvable in polynomial time. Our main focus is the algorithmic complexity of finding out how funnel-like a given DAG is. To this end, we study the NP-hard problem of computing the arc-deletion distance to a funnel of a given DAG. We develop efficient exact and approximation algorithms for the problem and test them on synthetic random graphs and real-world graphs.