New Results on Directed Edge Dominating Set

TL;DR

This paper introduces improved fixed-parameter tractable algorithms, polynomial kernels, and approximation algorithms for directed edge dominating set problems, and provides a comprehensive complexity classification on tournaments.

Contribution

It significantly advances algorithmic solutions and complexity understanding for the directed edge dominating set problem, especially on tournaments and specific parameterizations.

Findings

Improved FPT algorithms for (0,1)-dEDS and (1,1)-dEDS.

Polynomial kernels and constant-factor approximations for key cases.

Complete complexity classification for directed edge dominating set on tournaments.

Abstract

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.