Local Search Improvements for Soft Happy Colouring
Mohammad Hadi Shekarriz, Dhananjay Thiruvady, Asef Nazari, Wilfried Imrich

TL;DR
This paper introduces local search algorithms to improve soft happy colouring in graphs, demonstrating that higher happiness thresholds enhance community detection accuracy and that efficient algorithms can significantly increase the number of happy vertices.
Contribution
It proposes three local search algorithms for soft happy colouring, highlighting a fast, reliable linear-time method that improves the number of happy vertices and community detection.
Findings
Higher happiness thresholds improve community detection accuracy.
The linear-time local search algorithm is fast and significantly increases happy vertices.
Algorithms perform well compared to existing methods.
Abstract
For and a coloured graph , a vertex is -happy if at least of its neighbours have the same colour as . Soft happy colouring of a partially coloured graph is the problem of finding a vertex colouring that preserves the precolouring and has the maximum number of -happy vertices. It is already known that this problem is NP-hard and directly relates to the community structure of the graphs; under a certain condition on the proportion of happiness and for graphs with community structures, the induced colouring by communities can make all the vertices -happy. We show that when , a complete -happy colouring has a higher accuracy of community detection than a complete -happy colouring. Moreover, when is greater than a threshold, it is unlikely for an…
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