Graph Colouring

Predicates

• Input: node/1, link/2, colour/1

• Output: chosenColour/2

Problem Description

A graph is a set of nodes and a symmetric, binary link relation on nodes. Given a set of N colours, a graph is colourable if each node can be assigned a colour in such a way that any two nodes that are linked together cannot have the same colour.

Input format

A number of node facts which give the names of the nodes. Node names are consecutive, ascending integers starting from 1.

A number of colour facts which give the names of the colours. Colour names start with the sequence "red", "green", "blue".

A number of link facts which say which nodes are linked. Note that if link(N1,N2). is included then so will link(N2,N1).

For example:

node(1).
node(2).
node(3).
link(1,2).
link(2,1).
link(2,3).
link(3,2).
link(3,1).
link(1,3).
colour(red).
colour(green).
colour(blue).

Output format

The initial facts and a set of choosenColour predicates, one for each node, specifying the node's colour. Continuing the example:

chosenColour(1,red).
chosenColour(2,green).
chosenColour(3,blue).

Calibration

The Two Possible Values of the Chromatic Number of a Random Graph (with A. Naor)
Annals of Mathematics, 162 (3), (2005), 1333-1349.
http://www.cs.ucsc.edu/~optas/papers/kcol.pdf

Suggests that given a random graph with n nodes and a density of (d/n) then the chromatic number is either k or k+1 where k is the smallest number such that d < 2k log(k).

Thus settings with around 125-150 nodes (135 is good), link density of 0.1 (d = 12-15) and 5 colours gives difficult programs.

./graphGenerator.pl --nodes=$N --density=$D --colours=$C

Example

Comment

This problem was part of the Second ASP Competition and was proposed by Martin Brain.

Author(s)

• Author: Yuliya Lierler
  • Affiliation: University of Kentucky, USA
• Author: Marcello Balduccini
  • Affiliation: Kodak Research Labs, USA