Embedding a novel objective function in a two-phased local search for robust vertex coloring

In this paper, we propose a two-phased local search for vertex coloring. The algorithm alternately executes two closely interacting functionalities, i.e., a stochastic and a deterministic local search. The stochastic phase is basically based on biased random sampling that, according to a probability matrix storing the probability a vertex can be assigned to a color, iteratively constructs feasible colorings. The deterministic phase, instead, consists in assigning sequentially, according to a given ordering, each vertex to the color which causes the lowest increase of the solution penalty, and then, when the schedule is constructed, swap operations are executed to improve the performance. The interaction between the two phases is implemented by tunnelling information of what happened during a phase to the successive ones. Beyond the algorithm scheme, the novelty of the approach stems from the fact that the objective function is not minimizing the number of colors but a new penalty function. The proposed approach is tested on known benchmarks for the studied problem available on the public domain. From a comparison to the state of the art it appears that the proposed approach is robust and is able to achieve best known results.