The Complexity of Finding (Approximate Sized) Distance-d Dominating Set in Tournaments

Biswas, Arindam; Jayapaul, Varunkumar; Raman, Venkatesh; Satti, Srinivasa Rao

doi:10.1007/978-3-319-59605-1_3

Cited by 2 publications

(2 citation statements)

References 15 publications

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“…The generalization of Dominating Set where vertices dominate their r-neighborhood has also been well-studied in general [8,12,16,27,29]. This problem is much easier on tournaments for r ≥ 2, as the size of the solution is always a constant [5].…”

Section: Range Of P Qmentioning

confidence: 99%

See 1 more Smart Citation

New Results on Directed Edge Dominating Set

Belmonte¹,

Hanaka²,

Katsikarelis³

et al. 2019

Preprint

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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.

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Section: Range Of P Qmentioning

confidence: 99%

“…One such king is the vertex of maximum out-degree (see e.g. [5]). It is folklore that any tournament contains a Hamiltonian path, i.e.…”

Section: Complexity Backgroundmentioning

confidence: 99%

New Results on Directed Edge Dominating Set

Belmonte¹,

Hanaka²,

Katsikarelis³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

New Results on Directed Edge Dominating Set

Belmonte¹,

Hanaka²,

Katsikarelis³

et al. 2023

Discrete Mathematics &Amp; Theoretical Computer Science

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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.

show abstract

The Complexity of Finding (Approximate Sized) Distance-d Dominating Set in Tournaments

Cited by 2 publications

References 15 publications

New Results on Directed Edge Dominating Set

New Results on Directed Edge Dominating Set

New Results on Directed Edge Dominating Set

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