Map Coloring

Another fun challenge from Decision Management Community. A canonical example of the constraint programming (over finite domains) paradigm leads to a simple and elegant SQL solution.

The task is to color a map of six European countries: Belgium, Denmark, France, Germany, Luxembourg, and the Netherlands in such a way that no neighboring countries use the same color. You need to use no more than 4 colors: blue, red, green, or yellow.

Starting with two sets: country and color; who borders whom, and the rule.

country {BE, DK, FR, DE, LU, NL}
color   {B, R, G, Y}

BE borders {FR, LU, DE, NL}
DK borders {DE}
FR borders {BE, LU, DE}
DE borders {FR, LU, BE, NL, DK}
LU borders {FR, DE, BE}
NL borders {BE, DE}

Neighboring countries can not use the same color.

Reasoning

There are six countries, each one can have one of four colors, so in total there is a maximum of 4^6 = 4096 combinations to examine. The strategy is simple:

  1. Create a set of all possible combinations.
  2. Apply constraints.
  3. Present all results that satisfy given constraints.

The SQL code is straightforward and easy to follow. Try it, there are 144 possible solutions.

Variation

In a variation of the problem rules are changed a bit. Only three colors are allowed and some neighboring countries are allowed to have the same color, but there is a penalty cost assigned. The task is to find solutions with minimum penalty cost.

country {BE, DK, FR, DE, LU, NL}
color   {R, G, B}

These neighboring countries can use the same color,
with assigned cost:

LU-FR  $257
LU-DE  $904
LU-BE  $568

Reasoning v2

This time there are 3^6 = 729 combinations to examine. The strategy:

  1. Create a set of all possible combinations.
  2. Apply mandatory constraints.
  3. Calculate the cost for all tuples satisfying mandatory constraints.
  4. Find the minimum cost.
  5. Present all results that have the minimum cost.

Again, SQL code is easy to follow, this time there are 12 possible solutions.