Geodesic
New Results on Directed Edge Dominating Set
Discrete mathematics & theoretical computer science, Tome 25 (2023-2024) no. 1.

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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.
DOI : 10.46298/dmtcs.5378
Classification : 05C20, 05C69, 05C85, 68Q25, 68R10 
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     author = {Belmonte, R\'emy and Hanaka, Tesshu and Katsikarelis, Ioannis and Kim, Eun Jung and Lampis, Michael},
     title = {New {Results} on {Directed} {Edge} {Dominating} {Set}},
     journal = {Discrete mathematics & theoretical computer science},
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     year = {2023-2024},
     doi = {10.46298/dmtcs.5378},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.5378/}
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Belmonte, Rémy; Hanaka, Tesshu; Katsikarelis, Ioannis; Kim, Eun Jung; Lampis, Michael. New Results on Directed Edge Dominating Set. Discrete mathematics & theoretical computer science, Tome 25 (2023-2024) no. 1. doi : 10.46298/dmtcs.5378. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.5378/

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