New Results on Directed Edge Dominating Set
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 just by tw, 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.
Edge Dominating Set
Tournaments
Treewidth
Theory of computation~Parameterized complexity and exact algorithms
67:1-67:16
Regular Paper
Partially supported by JSPS and MAEDI under the Japan-France Integrated Action Program (SAKURA), Project GRAPA 38593YJ.
Rémy
Belmonte
Rémy Belmonte
University of Electro-Communications, Chofu, Tokyo, 182-8585, Japan
Tesshu
Hanaka
Tesshu Hanaka
Department of Information and System Engineering, Chuo University, Tokyo, Japan
Ioannis
Katsikarelis
Ioannis Katsikarelis
Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, 75016, Paris, France
Eun Jung
Kim
Eun Jung Kim
Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243 , LAMSADE, 75016, Paris, France
The author was partially supported by the ANR grant "ESIGMA" (ANR-17-CE40-0028).
Michael
Lampis
Michael Lampis
Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243 , LAMSADE, 75016, Paris, France
The author was partially supported by the ANR grant "ESIGMA" (ANR-17-CE40-0028).
10.4230/LIPIcs.MFCS.2018.67
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Rémy Belmonte, Tesshu Hanaka, Ioannis Katsikarelis, Eun Jung Kim, and Michael Lampis
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