Abstract
In this paper we consider a variant of graph colouring known as the maximum happy vertices problem. This problem involves taking a graph in which a subset of the vertices have been preassigned to colours. The objective is to then colour the remaining vertices such that the number of happy vertices is maximised, where a vertex is considered happy only when it is assigned to the same colour as all of its neighbours. We design and test a tabu search approach, which is compared to two existing state of the art methods. We see that this new approach is particularly suited to larger problem instances and finds very good solutions in very short time frames. We also propose a algorithm to find upper bounds for the problem efficiently. Moreover, we propose an algorithm for imposing additional precoloured vertices and are hence able to significantly reduce the solution space. Finally, we present an analysis of this problem and use probabilistic arguments to characterise problem hardness.
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Notes
Note that when the desired number of precoloured vertices \(|V'|\) is less than the number of available colours k, instances cannot be generated. As a result, such parameter combinations are not present in our analyses.
Methods were implemented in C++ and compiled with GCC-5.4.0. Our code is available at http://www.rhydlewis.eu/resources/happytabu.zip. The experiments were conducted on Monash University’s Campus Cluster, where each machine in the cluster consists of 24 cores and 256 GB RAM. Each physical core has two hyper-threaded cores with Intel Xeon E5-2680 v3 2.5GHz, 30M Cache, 9.60GT/s QPI, Turbo, HT, 12C/24T (120W). For tabu search a setting of \(\tau = 2\) was used.
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Thiruvady, D., Lewis, R. & Morgan, K. Tackling the maximum happy vertices problem in large networks. 4OR-Q J Oper Res 18, 507–527 (2020). https://doi.org/10.1007/s10288-020-00431-4
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DOI: https://doi.org/10.1007/s10288-020-00431-4