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New Results on Directed Edge Dominating Set

Authors Rémy Belmonte, Tesshu Hanaka, Ioannis Katsikarelis, Eun Jung Kim, Michael Lampis



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Rémy Belmonte
  • University of Electro-Communications, Chofu, Tokyo, 182-8585, Japan
Tesshu Hanaka
  • Department of Information and System Engineering, Chuo University, Tokyo, Japan
Ioannis Katsikarelis
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, 75016, Paris, France
Eun Jung Kim
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243 , LAMSADE, 75016, Paris, France
Michael Lampis
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243 , LAMSADE, 75016, Paris, France

Cite AsGet BibTex

Rémy Belmonte, Tesshu Hanaka, Ioannis Katsikarelis, Eun Jung Kim, and Michael Lampis. New Results on Directed Edge Dominating Set. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 67:1-67:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.MFCS.2018.67

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 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Edge Dominating Set
  • Tournaments
  • Treewidth

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