Local Search Improvements for Soft Happy Colouring

Abstract: For $0\leq \rho\leq 1$ and a coloured graph $G$, a vertex $v$ is $\rho$-happy if at least $\rho \mathrm{deg}(v)$ of its neighbours have the same colour as $v$. Soft happy colouring of a partially coloured graph $G$ is the problem of finding a vertex colouring $\sigma$ that preserves the precolouring and has the maximum number of $\rho$-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 $\rho$ and for graphs with community structures, the induced colouring by communities can make all the vertices $\rho$-happy. We show that when $0\leq \rho_1<\rho_2\leq 1$, a complete $\rho_2$-happy colouring has a higher accuracy of community detection than a complete $\rho_1$-happy colouring. Moreover, when $\rho$ is greater than a threshold, it is unlikely for an algorithm to find a complete $\rho$-happy colouring with colour classes of almost equal sizes. Three local search algorithms for soft happy colouring are proposed, and their performances are compared with one another and other known algorithms. Among them, the linear-time local search is shown to be not only very fast, but also a reliable algorithm that can dramatically improve the number of $\rho$-happy vertices.