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.
Nowhere dense graph classes and algorithmic applications. A tutorial at Highlights of Logic, Games and Automata 2019
无处稠密图类的概念是由Nešetzil和Ossona de Mendez提出的，它提供了一个鲁棒的图类一致稀疏的概念。无处密集类推广了许多熟悉的类的稀疏图，如类，排除一个固定的图作为一个小的或拓扑的小。他们承认几个看似无关的自然特征，导致强大的算法应用。特别是，一阶逻辑的模型检查问题是固定参数的这些类的易处理的。这些笔记是为2019年逻辑，游戏和自动机亮点的教程准备的，是对算法应用驱动的无处密集理论的简要介绍。
Algorithms and Hardness for Geodetic Set on Tree-like Digraphs
Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS（克莱蒙奥弗涅大学、法国国家科学研究中心、克莱蒙奥弗涅理工大学、圣艾蒂安国立高等矿业学校、信息与系统优化实验室）；School of Mathematics, Statistics, and Computer Science, University of Tehran（德黑兰大学数学、统计与计算机科学学院）；Indian Institute of Science Education and Research Pune（浦那科学教育与研究所）
Algorithms to Approximate Column-Sparse Packing Problems
列稀疏填充问题出现在确定性和随机离散优化的几个上下文中。我们提出了两个统一的想法，（非均匀）衰减和多机会算法，以获得改进的近似算法的一些著名的家庭这样的问题。作为三个主要的例子，我们获得了k列稀疏填充整数规划的已知LP松弛的完整性间隙，直到低阶项（Bansal等人，Theory of Computing，2012）和随机k-集合包装（Bansal等人，Furedi，Kahn和Seymour关于超图匹配的一个主要的完整性间隙猜想（Combinatorica，1993）。
