Algorithms and Experiments on Colouring Squares of Planar Graphs

In this work we study the important problem of colouring squares of planar graphs (SQPG). We design and implement two new algorithms that colour in a different way SQPG. We call these algorithms MDsatur and RC. We have also implemented and experimentally evaluated the performance of most of the known approximation colouring algorithms for SQPG [14, 6, 4, 10]. We compare the quality of the colourings achieved by these algorithms, with the colourings obtained by our algorithms and with the results obtained from two well-known greedy colouring heuristics. The heuristics are mainly used for comparison reasons and unexpectedly give very good results. Our algorithm MDsatur outperforms the known algorithms as shown by the extensive experiments we have carried out. The planar graph instances whose squares are used in our experiments are “non-extremal” graphs obtained by LEDA and hard colourable graph instances that we construct. The most interesting conclusions of our experimental study are: 1) all colouring algorithms considered here have almost optimal performance on the squares of “non-extremal” planar graphs. 2) all known colouring algorithms especially designed for colouring SQPG, give significantly better results, even on hard to colour graphs, when the vertices of the input graph are randomly named. On the other hand, the performance of our algorithm, MDsatur, becomes worse in this case, however it still has the best performance compared to the others. MDsatur colours the tested graphs with 1.1 OPT colours in most of the cases, even on hard instances, where OPT denotes the number of colours in an optimal colouring. 3) we construct worst case instances for the algorithm of Fotakis el al. [6], which show that its theoretical analysis is tight.