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Now, in order to translate this into a CNF SAT problem, I understand that:

  1. Each node must have atleast one color.
  2. Each node must have at most one color.
  3. Connected nodes cannot have the same color.

But I can't move ahead as I am unable to understand what to return. If we have a function Convert(graph) where the graph has 3 nodes namely nodes = (0,1,2) and 3 edges namely edges = [(0,1), (0,2), (1,2)] with possible colors = (1,2,3); what should the output for the conversion into a CNF SAT Form for this particular case look like? Thank you

Silent
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1 Answers1

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Here is a simple direct encoding:

For every node, introduce k variables (k = number of colors), and a clause that simply lists those variables (forcing at least one color to be chosen).

For every edge (X, Y) and color c, introduce a binary clause that prevents X and Y from both having color c at the same time, ie -Xc -Yc.

For this graph, the total CNF in DIMACS format could look like this:

p cnf 9 12
1 2 3 0
4 5 6 0
7 8 9 0
-1 -4 0
-2 -5 0
-3 -6 0
-4 -7 0
-5 -8 0
-6 -9 0
-1 -7 0
-2 -8 0
-3 -9 0

A solver might give (for example, and obviously there are more possibilities) 1 -2 -3 -4 -5 6 -7 8 -9 0 meaning that node 0 got color 1, node 1 got color 3, and node 2 got color 2.

By the way note that the constraint "each node must have at most one color" was not used. Trivial post-processing can deal with that. For some graphs and some numbers of colors you may get a result where a node is assigned multiple colors (only if it does not share any of those colors with a neighbour, and usually if it's possible it still won't happen because the solver has no incentive to do it), you can simply pick any of them. If that's not acceptable, add a binary clause for every pair of variables associated with a node to ban them from being both set, eg -1 -2 0, -1 -3 0 and so on.

There are more advanced encodings that scale better to larger graphs.

harold
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