> that instance is too hard, so we make it easier by forcing a color in the center (this is a form of symmetry breaking)
The example of symmetry breaking in your post is "the instance of the largest color that appears closest to the center must appear in octant II".
I can understand this - by using a combination of rotations and reflections, any coloring that doesn't satisfy this constraint can be transformed into one that does. So the patterns we reject by using this "symmetry-breaking" approach are those that are identical, through one or more symmetries, to a pattern we won't reject. So far so good.
But forcing a color in the center can't be symmetry-breaking in this sense. Every symmetry of the diamond will map the center onto itself, so applying a "symmetry-breaking" rule that the center must be painted a particular color will let you rule out a total of zero patterns that can be transformed into patterns that match your rule. That is, literally, worthless.
Such a constraint will let you rule out patterns that aren't transformable into the patterns you do consider... but that's bad, not good.
What do you mean when you say that forcing a color in the center is a form of symmetry breaking?
The example of symmetry breaking in your post is "the instance of the largest color that appears closest to the center must appear in octant II".
I can understand this - by using a combination of rotations and reflections, any coloring that doesn't satisfy this constraint can be transformed into one that does. So the patterns we reject by using this "symmetry-breaking" approach are those that are identical, through one or more symmetries, to a pattern we won't reject. So far so good.
But forcing a color in the center can't be symmetry-breaking in this sense. Every symmetry of the diamond will map the center onto itself, so applying a "symmetry-breaking" rule that the center must be painted a particular color will let you rule out a total of zero patterns that can be transformed into patterns that match your rule. That is, literally, worthless.
Such a constraint will let you rule out patterns that aren't transformable into the patterns you do consider... but that's bad, not good.
What do you mean when you say that forcing a color in the center is a form of symmetry breaking?