Finding happiness by evolutionary algorithms

TL;DR

This paper tackles Soft Happy Colouring (SHC), an NP-hard problem linking graph colouring to community structure. It introduces a memetic algorithm that hybrids genetic operators with fast local search, evaluated on stochastic block model graphs. The results show that the memetic approach, especially MA+RLS(LS), yields the strongest SHC performance, while MA(LMC) excels at recovering communities under favorable ρ ranges. Overall, the study demonstrates the value of combining fast heuristics with evolutionary search for SHC and suggests directions for exact approaches to better delineate SHC complexity.

Abstract

A recent line of research concerns the problem of soft happy colouring (SHC), which requires that a partially coloured graph be extended to a complete colouring to maximise local agreements, so that as many vertices as possible end up surrounded by enough same-coloured neighbours. It is already known that SHC is NP-hard, and its solutions have a direct relationship with the community structure of networks; thus, it has immense applications in security and resilience. Past studies have shown that local search approaches can be fast and effective to an extent on the SHC; however, they often get stuck in local optima. Regarding the related problem of maximising happy vertices, evolutionary approaches have been proven effective; hence, this study develops a customised memetic algorithm that is a hybrid of genetic algorithms and local search. The experimental evaluation on a range of graphs in the stochastic block model shows that the memetic algorithm can achieve excellent results in search for an optimised solution to SHC compared to the local search approaches and standard genetic algorithms. Moreover, learning and evolution in the memetic algorithm allow diversification of solutions generated by fast, effective local search approaches, which prove superior for the challenging problem of community detection.

Figures (15)

• Figure 1: A connected graph $G$ with 11 communities. Grey lines are inter-community edges. The colouring $\sigma$ here is aligned with the communities and is a complete 0.6-happy colouring. The objective of community detection is to find such a vertex partition.
• Figure 2: The graph depicted on the left (a) is being coloured by LMC. Vertices denoted by $r$, $g$ and $b$ are precoloured vertices. At this stage, the vertex $v$ is selected to be coloured. Because it has two red, one blue, one green, and one uncoloured neighbours, the colour red is given to it (b).
• Figure 3: The number of $\rho$-happy vertices in the coloured graph of (a) is getting improved by LS, and the free vertex $v$ is selected. Because it has three green, two red, and one blue neighbours, its colour is changed from red to green (b).
• Figure 4: This is a flowchart of EAs. The improvement step (pale pink dashed-rectangle) is essential in MAs, but it is skipped in GAs.
• Figure 5: The procedure CrossOver starts by selecting two solutions (parents). At the beginning (stage 0), the free vertices of the offspring are uncoloured. During CrossOver (stage 1), some free vertices are randomly selected from the first parent and other free vertices are selected from the second parent. Then the colours of the selected vertices are transferred to the free vertices of the offspring.