In this problem, three colors are not going to be enough to color the map in such a way that no neighboring countries are the same color. So, what is the best approach for dealing with this situation? One option is to select what countries will share a color. If you look at the map, you could guess that if you removed one country, for example Luxembourg, from the set of constraints that no neighboring countries are the same color, you might be able to find a solution using just three colors. This is known as relaxing constraints.

However, you might want to try to have Luxembourg be a different color from some its neighbors, even if it cannot be a different color from all its neighbors. Suppose that you would most like to have Germany and Luxembourg be different colors if possible, then Belgium and Luxembourg if possible, and then France and Luxembourg if possible. You can represent these wishes as an objective. The relative importance of different constraints can be expressed in an objective. You can also use an objective to select the most appropriate solution when a problem has more than one solution--if there is more than one set of values for the variables that satisfy the constraints of the problem.

Write a natural language description of this problem. Answer these questions: