Map Colouring

Map colouring is a puzzle where you assign different colours to different features ( in this case counties ) on a map with the goal of having no 2 colours share a border. It was first introducted in 1852 to determine the minimum number of colors when trying to colour a map of England. It was postulated that 4 colours were the minimum number of colours to fill a map so that no regions sharing a common border had the same colour.

This problem is called the Four color theorem and was the first major theorem to be proved using a computer in 1976.

You can represent the map as a graph, with the counties as nodes and the borders as connections. This type of problem can be gerneraled and used in other problems such as scheduling, register allocation and pattern matching.

Solving

As I go through the Discrete Optimization course, I'm learning new tecniques to use and get an understanding of the working space fo the problem. Like the other problems in this course, this is an NP problem.

Brute force

I would start by trying every single combination and returning the best solution. Computers are great with numbers and I want to be lazy. But looking at the table above there is no way to complete this in a reasonable time.

Name Nodes Colours Number of States Time to Solve*
Small Example 20 3 3,486,784,401 or 10^9 2 seconds
Counties of England 83 4 10^50 10^36 years
Countries of the World 195 4 10^117 10^103 years

*Assuming 2GHz computer, very optimistically checking 1 combination a cycle

We need another way

Pick most connected

Instead of trying every combination, lets order the way we select the nodes we colour. Starting from the node which has the most connections, choosing its colour and moving to the next, adding a new colour if neighbouring nodes used all the previous colors. This will run until all the colours are chosen

This is so much faster! It is a greedy algorithm so it doesn’t give us an optimal result, but the speed is important here. When you add a new node, it adds only a small amount of time and is close to linear.

Saturation

This method is very similar to the “most connected” but in this case the next node to select will update each time. When a node is colored the connected nodes are updated with a new saturation number. This saturation number tells how many different colors a node neighbours with. The other name for this algorithm is DSATUR and is a well known algorithm for completing this problem

Next...

This post is a work in progress and I hope to return. So far it has been interesting on how large NP problems can get and we have to try different algorithms. A good enough solution is much easier to get than a perfect solution. I will continue to write more about this as I go through the class and it looks like there is some possibility of using local search.

Thank you for reading