Algorithmic Properties of Sparse Digraphs

TL;DR

This paper explores the properties of sparse directed graphs, extending concepts from undirected graph theory, and demonstrates new fixed-parameter tractability and approximation results for key problems like Steiner tree and dominating set.

Contribution

It introduces directed bounded expansion and nowhere crownfulness, showing that many algorithmic tools from undirected graphs apply to directed graphs, and provides new FPT and approximation algorithms.

Findings

Directed Steiner tree problem is fixed-parameter tractable on bounded expansion classes.

Distance-$r$ dominating set can be approximated within an $O(\log k)$ factor.

Polynomial kernels are achievable for distance-$r$ dominating sets on nowhere crownful classes.

Abstract

The notions of bounded expansion and nowhere denseness have been applied very successfully in algorithmic graph theory. We study the corresponding notions of directed bounded expansion and nowhere crownfulness on directed graphs. We show that many of the algorithmic tools that were developed for undirected bounded expansion classes can, with some care, also be applied in their directed counterparts, and thereby we highlight a rich algorithmic structure theory of directed bounded expansion classes. More specifically, we show that the directed Steiner tree problem is fixed-parameter tractable on any class of directed bounded expansion parameterized by the number $k$ of non-terminals plus the maximal diameter $s$ of a strongly connected component in the subgraph induced by the terminals. Our result strongly generalizes a result of Jones et al., who proved that the problem is fixed parameter tractable on digraphs of bounded degeneracy if the set of terminals is required to be acyclic. We furthermore prove that for every integer $r\geq 1$, the distance-$r$ dominating set problem can be approximated up to a factor $O(\log k)$ and the connected distance-$r$ dominating set problem can be approximated up to a factor $O(k\cdot \log k)$ on any class of directed bounded expansion, where $k$ denotes the size of an optimal solution. If furthermore, the class is nowhere crownful, we are able to compute a polynomial kernel for distance-$r$ dominating sets. Polynomial kernels for this problem were not known to exist on any other existing digraph measure for sparse classes.