We study a family of generalizations of Edge Dominating Set on directed graphs called Directed $(p,q)$-Edge Dominating Set. In this problem an arc $(u,v)$ is said to dominate itself, as well as all arcs which are at distance at most $q$ from $v$, or at distance at most $p$ to $u$. First, we give significantly improved FPT algorithms for the two most important cases of the problem, $(0,1)$-dEDS and $(1,1)$-dEDS (that correspond to versions of Dominating Set on line graphs), as well as polynomial kernels. We also improve the best-known approximation for these cases from logarithmic to constant. In addition, we show that $(p,q)$-dEDS is FPT parameterized by $p+q+tw$, but W-hard parameterized by $tw$ (even if the size of the optimal is added as a second parameter), where $tw$ is the treewidth of the underlying graph of the input. We then go on to focus on the complexity of the problem on tournaments. Here, we provide a complete classification for every possible fixed value of $p,q$, which shows that the problem exhibits a surprising behavior, including cases which are in P; cases which are solvable in quasi-polynomial time but not in P; and a single case $(p=q=1)$ which is NP-hard (under randomized reductions) and cannot be solved in sub-exponential time, under standard assumptions.

翻译：我们研究有向图中边支配集的一类推广问题，称为有向$(p,q)$-边支配集。在该问题中，一条弧$(u,v)$被认为支配自身，以及所有距离$v$不超过$q$或距离$u$不超过$p$的弧。首先，我们针对该问题的两个最重要情况——$(0,1)$-dEDS和$(1,1)$-dEDS（对应于线图上支配集的版本），给出了显著改进的FPT算法以及多项式核。我们还将这些情况的最佳近似比从对数改进为常数。此外，我们证明$(p,q)$-dEDS在以$p+q+tw$为参数时是FPT的，但在以$tw$为参数时是W-困难的（即使将最优解大小作为第二个参数），其中$tw$是输入底层图的树宽。接着，我们聚焦于该问题在竞赛图上的复杂度。我们针对每个可能的固定$p,q$值给出了完整分类，结果表明该问题展现出令人惊讶的行为，包括在P中的情况、可在拟多项式时间内求解但不在P中的情况，以及一个单独情况$(p=q=1)$，该情况在标准假设下是NP-困难的（随机归约下）且无法在次指数时间内求解。