Weighted coloring on planar, bipartite and split graphs: complexity and improved approximation
Résumé
We study complexity and approximation of min weighted node coloring in natural graph-families as planar, bipartite and split graphs. We show that this problem is NP-complete in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-complete in P_8-free bipartite graphs, but it turns to be polynomial for P_5-free ones. We next focus ourselves on approximability in general bipartite graphs and improve older approximation results by approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. Improving, here also, older results, we show that this problem remains strongly NP-complete, even in the case where the input-graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7/6 -epsilon, for any epsilon > 0 and we match it with an approximation ratio of the same value. Finally, we deal with approximation of min weighted node coloring in split graphs and show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.