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- W4285597663 abstract "We present fixed-parameter tractable (FPT) algorithms for two problems, Maximum Happy Set (MaxHS) and Maximum Edge Happy Set (MaxEHS)--also known as Densest k-Subgraph. Given a graph $G$ and an integer $k$, MaxHS asks for a set $S$ of $k$ vertices such that the number of $textit{happy vertices}$ with respect to $S$ is maximized, where a vertex $v$ is happy if $v$ and all its neighbors are in $S$. We show that MaxHS can be solved in time $mathcal{O}left(2^textsf{mw} cdot textsf{mw} cdot k^2 cdot |V(G)|right)$ and $mathcal{O}left(8^textsf{cw} cdot k^2 cdot |V(G)|right)$, where $textsf{mw}$ and $textsf{cw}$ denote the $textit{modular-width}$ and the $textit{clique-width}$ of $G$, respectively. This resolves the open questions posed in literature. The MaxEHS problem is an edge-variant of MaxHS, where we maximize the number of $textit{happy edges}$, the edges whose endpoints are in $S$. In this paper we show that MaxEHS can be solved in time $f(textsf{nd})cdot|V(G)|^{mathcal{O}(1)}$ and $mathcal{O}left(2^{textsf{cd}}cdot k^2 cdot |V(G)|right)$, where $textsf{nd}$ and $textsf{cd}$ denote the $textit{neighborhood diversity}$ and the $textit{cluster deletion number}$ of $G$, respectively, and $f$ is some computable function. This result implies that MaxEHS is also fixed-parameter tractable by $textit{twin cover number}$." @default.
- W4285597663 created "2022-07-16" @default.
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- W4285597663 date "2022-07-13" @default.
- W4285597663 modified "2023-09-27" @default.
- W4285597663 title "Improved Parameterized Complexity of Happy Set Problems" @default.
- W4285597663 doi "https://doi.org/10.48550/arxiv.2207.06623" @default.
- W4285597663 hasPublicationYear "2022" @default.
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