Kernelization for Maximum Happy Vertices Problem

  • Hang Gao
  • Wenyu Gao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


The homophyly phenomenon is very common in social networks. The Maximum Happy Vertices (MHV) is a newly proposed problem related to homophyly phenomenon. Given a graph \(G=(V,E)\) and a vertex coloring of G, we say that a vertex v is happy if v shares the same color with all its neighbors, and unhappy, otherwise, and that an edge e is happy, if its two endpoints have the same color, and unhappy, otherwise. Given a partial vertex coloring of G with k number of different colors, the k-MHV problem is to color all the remaining vertices such that the number of happy vertices is at least l. We study k-MHV from the parameterized algorithm perspective; we prove that k-MHV has an exponential kernel of \(2^{kl+l}\,+\,kl\,+\,k\,+\,l\) on general graph. For planar graph, we get a much better polynomial kernel of \(7(kl+l)+k-10\).


Maximum happy vertices Happy coloring Parameterized complexity Kernelization Planar graph 



We would like to thank the anonymous reviewers for their detailed reviews and suggestions.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of TransportationJilin UniversityChangchunChina
  2. 2.School of Information ScienceGuangdong University of Finance and EconomicsGuangzhouChina

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