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New Results on Directed Edge Dominating Set

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Titel: New Results on Directed Edge Dominating Set
Autoren: Belmonte, Rémy, Hanaka, Tesshu, Katsikarelis, Ioannis, Kim, Eun Jung, Lampis, Michael
Weitere Verfasser: Rémy Belmonte and Tesshu Hanaka and Ioannis Katsikarelis and Eun Jung Kim and Michael Lampis
Quelle: Discrete Mathematics & Theoretical Computer Science, Vol vol. 25:1 (2023)
Publication Status: Preprint
Verlagsinformationen: Centre pour la Communication Scientifique Directe (CCSD), 2023.
Publikationsjahr: 2023
Schlagwörter: FOS: Computer and information sciences, Programmation, Computational Complexity, Data Structures and Algorithms, Treewidth, Tournaments, 0102 computer and information sciences, Computational Complexity (cs.CC), Edge Dominating Set, 01 natural sciences, organisation des données, computer science - data structures and algorithms, Computer Science - Computational Complexity, Computer Science - Data Structures and Algorithms, QA1-939, Data Structures and Algorithms (cs.DS), computer science - computational complexity, ddc:004, Mathematics, logiciels
Beschreibung: 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.
Publikationsart: Article
Conference object
Dateibeschreibung: application/pdf
Sprache: English
ISSN: 1365-8050
DOI: 10.46298/dmtcs.5378
DOI: 10.48550/arxiv.1902.04919
DOI: 10.4230/lipics.mfcs.2018.67
Zugangs-URL: http://arxiv.org/abs/1902.04919
https://dmtcs.episciences.org/5378
https://doi.org/10.46298/dmtcs.5378
https://doaj.org/article/804a99ed163d4f95b99b4ae8a732ba86
https://dblp.uni-trier.de/db/journals/corr/corr1902.html#abs-1902-04919
https://doi.org/10.4230/LIPIcs.MFCS.2018.67
https://drops.dagstuhl.de/opus/volltexte/2018/9649/
https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.67
Rights: arXiv Non-Exclusive Distribution
CC BY
URL: https://arxiv.org/licenses/nonexclusive-distrib/1.0
Dokumentencode: edsair.doi.dedup.....0f788ea9d700a91fdffdbf53e6dec0fa
Datenbank: OpenAIRE
Beschreibung
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.
ISSN:13658050
DOI:10.46298/dmtcs.5378