Parameterized Complexity of Maximum Happy Set and Densest k-Subgraph

Authors Yosuke Mizutani , Blair D. Sullivan

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Yosuke Mizutani
  • School of Computing, University of Utah, Salt Lake City, UT, USA
Blair D. Sullivan
  • School of Computing, University of Utah, Salt Lake City, UT, USA


We thank Arnab Banerjee, Oliver Flatt and Thanh Son Nguyen for their contributions to a course project that led to this research.

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Yosuke Mizutani and Blair D. Sullivan. Parameterized Complexity of Maximum Happy Set and Densest k-Subgraph. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We present fixed-parameter tractable (FPT) algorithms for two problems, Maximum Happy Set (MaxHS) and Densest k-Subgraph (DkS) - also known as Maximum Edge Happy Set. 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^mw ⋅ mw ⋅ k² ⋅ |V(G)|) and 𝒪(8^cw ⋅ k² ⋅ |V(G)|), where mw and cw denote the modular-width and the clique-width of G, respectively. This answers the open questions on fixed-parameter tractability posed in [Asahiro et al., 2021]. The DkS problem asks for a subgraph with k vertices maximizing the number of edges. If we define happy edges as the edges whose endpoints are in S, then DkS can be seen as an edge-variant of MaxHS. In this paper we show that DkS can be solved in time f(nd)⋅|V(G)|^𝒪(1) and 𝒪(2^{cd}⋅ k² ⋅ |V(G)|), where nd and cd denote the neighborhood diversity and the cluster deletion number of G, respectively, and f is some computable function. This result implies that DkS is also fixed-parameter tractable by twin cover number.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Fixed parameter tractability
  • parameterized algorithms
  • maximum happy set
  • densest k-subgraph
  • modular-width
  • clique-width
  • neighborhood diversity
  • cluster deletion number
  • twin cover


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