The robust coloring problem

Some problems can be modeled as graph coloring ones for which the criterion of minimizing the number of used colors is replaced by another criterion maintaining the number of colors as a constraint. Some examples of these problem types are introduced; it would be the case, for instance, of the problem of scheduling the courses at a university with a fixed number of time slots – the colors – and with the objective of minimizing the probability to include an edge to the graph with its endpoints equally colored. Based on this example, the new coloring problem introduced in this paper will be denoted as the Robust coloring problem, RCP for short. It is proved that this optimization problem is NP-hard and, consequently, only small-size problems could be solved with exact algorithms based on mathematical programming models; otherwise, for large size problems, some heuristics are needed in order to obtain appropriate solutions. A genetic algorithm which solves the RCP is outlined.