New Algorithms for Mixed Dominating Set
A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle the problem's complexity parame...
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| Published in: | Discrete mathematics and theoretical computer science Vol. 23 no. 1; no. Discrete Algorithms |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
DMTCS
30.04.2021
Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| ISSN: | 1365-8050, 1462-7264, 1365-8050 |
| Online Access: | Get full text |
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| Summary: | A mixed dominating set is a collection of vertices and edges that dominates
all vertices and edges of a graph. We study the complexity of exact and
parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open
questions. In particular, we settle the problem's complexity parameterized by
treewidth and pathwidth by giving an algorithm running in time $O^*(5^{tw})$
(improving the current best $O^*(6^{tw})$), as well as a lower bound showing
that our algorithm cannot be improved under the Strong Exponential Time
Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound
of $O^*((2 - \varepsilon)^{pw})$). Furthermore, by using a simple but so far
overlooked observation on the structure of minimal solutions, we obtain
branching algorithms which improve both the best known FPT algorithm for this
problem, from $O^*(4.172^k)$ to $O^*(3.510^k)$, and the best known
exponential-time exact algorithm, from $O^*(2^n)$ and exponential space, to
$O^*(1.912^n)$ and polynomial space. |
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| ISSN: | 1365-8050 1462-7264 1365-8050 |
| DOI: | 10.46298/dmtcs.6824 |