Improved Parameterized Complexity of Happy Set Problems
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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 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 𝒪(2^·· k^2 · |V(G)|) and 𝒪(8^· k^2 · |V(G)|), where and denote the modular-width and the 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 happy edges, the edges whose endpoints are in S. In this paper we show that MaxEHS can be solved in time f()·|V(G)|^𝒪(1) and 𝒪(2^· k^2 · |V(G)|), where and denote the neighborhood diversity and the cluster deletion number of G, respectively, and f is some computable function. This result implies that MaxEHS is also fixed-parameter tractable by twin cover number.
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