Abstract
Given an undirected graph \(G = (V, E)\) with \(|V| = n\) and a vertex coloring, a vertex v is happy if v and all its neighbors have the same color. An edge is happy if its end vertices have the same color. Given a partial coloring of the vertices of the graph using k colors, the Maximum Happy Vertices (also called k-MHV) problem asks to color the remaining vertices such that the number of happy vertices is maximized. The Maximum Happy Edges (also called k-MHE) problem asks to color the remaining vertices such that the number of happy edges is maximized. For arbitrary graphs, k-MHV and k-MHE are NP-Hard for \(k \ge 3\). In this paper we study these problems for trees. For a fixed k we present linear time algorithms for both the problems. In general, for any k the proposed algorithms take \(O(nk \log k)\) and O(nk) time respectively.
A.S. Kare—Faculty member of University of Hyderabad. This work is carried out as part of his PhD program at IIT Hyderabad.
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We thank the anonymous reviewers for their detailed reviews and suggestions.
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Aravind, N.R., Kalyanasundaram, S., Kare, A.S. (2016). Linear Time Algorithms for Happy Vertex Coloring Problems for Trees. In: Mäkinen, V., Puglisi, S., Salmela, L. (eds) Combinatorial Algorithms. IWOCA 2016. Lecture Notes in Computer Science(), vol 9843. Springer, Cham. https://doi.org/10.1007/978-3-319-44543-4_22
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DOI: https://doi.org/10.1007/978-3-319-44543-4_22
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