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The problem is said to contain symmetry if there exist classes of equivalent solutions — solutions, which are called symmetrical because there exist a simple mechanical procedure to obtain one from another. Graph Coloring Problem has a very obvious symmetry — in every solution we can freely swap colors, e.g. every red node repaint as blue, and every blue node repaint as red. Solutions of this kind aren't bad, just redundant, leading to much bigger search space. Symmetry breaking prunes the search space by removing symmetries from the problem.

- Problem: Same as before
- Assignment:
- Download and extract archive.
- Look at and comprehend
`csp_coloring.mzn`

model. - Try to solve the
`csp_coloring_data.dzn`

instance.- You can use model created during previous classes
- There is a chance, that problem would to difficult to be solved in a reasonable time.

- File
`csp_coloring_data2.dzn`

includes info about the largest clique in the graph`minColorsNumber`

- size of the largest clique`maxClique`

- indexes of the vertices forming the largest clique

- Improve model to make use of the info about the largest clique
- Try to solve the problem again.

- Definition: Knapsack problem with several identical knapsacks instead of one.
- Assignment:
- Find a symmetry in the problem.
- Download and extract the archive
- Look at and comprehend
`csp_n_knapsack.mzn`

.- Note : this model is not an example of good constraint model; it's just used to show a common technique to break symmetries
- Note use of the
`at_most`

global constraint. - There is a defined predicate named
`knapsack`

→ it's the first example of user defined predicate

- Run model with the associated data file:
- The problem may be to hard for the solver

- Break symmetry using ''lex_less'' global constraint.
- Run new model with the same data file

There is a good chance the problem can be defined in more than a one way. Also you may find a set of constraints that is sufficient to define the problem. That's cool, however there can exist so called “redundant constraints”; redundant because they do not have impact on the number or quality of the solutions. The only reason to include them into the model is that they may contain additional info about the structure of the problem, therefore giving solver an opportunity to prune the search space (most of the solver prune the search space by propagating constraints, redundant constraint may boost this process).

- Definition: Same as before
- Assignment:
- Download, extract, comprehend the model
- Add redundant constraints, hints:
- what should be equal the sum of the magic sequence?
- what should be equal the sum of sequence elements multiplied by their indexes (if indexing starts from 0)?

- Compare solving time with and without the redundant constraints.
- Smile with satisfaction

If you have more than one model of the same problem, you can combine them into a single model. Why would one do that? Mostly because some constraints are easier to express with different variables. Other reason could be that the second model often makes a great example of the redundant constraints.

- Problem: N-Queens again
- Assignment:
- Download, extract and comprehend the model (or use your own)
- Add another model of the problem
- try to use the boolean array of variables
`array[1..n, 1..n] of var bool: qb;`

(queen boolean) - add missing constraints so the second model was also independent

- Channel constraints from the both models:
- create constraint that connects variables from the model

- Compare running time of the normal and channeled model
- Add symmetry breaking to the problem by using
`lex_lesseq`

constraint on the different permutations of the`qb`

array- below the assignments there is a code listing with all permutations calculated in MiniZinc, can you tell what symmetries they represent?

- Compare running time again
- Give yourself a self-five, in this case, it may not improve the running time, but the technique itself is very useful in more complex problems

array[int] of var bool: qb0 = array1d(qb); array[int] of var bool: qb1 = [ qb[j,i] | i,j in 1..n ]; array[int] of var bool: qb2 = [ qb[i,j] | i in reverse(1..n), j in 1..n ]; array[int] of var bool: qb3 = [ qb[j,i] | i in 1..n, j in reverse(1..n) ]; array[int] of var bool: qb4 = [ qb[i,j] | i in 1..n, j in reverse(1..n) ]; array[int] of var bool: qb5 = [ qb[j,i] | i in reverse(1..n), j in 1..n ]; array[int] of var bool: qb6 = [ qb[i,j] | i,j in reverse(1..n) ]; array[int] of var bool: qb7 = [ qb[j,i] | i,j in reverse(1..n) ];

Reification in Constraint Programming means treating the constraint as a first-order citizen, i.e. you can use the constraint as a boolean value in your model. If you've used the `bool2int`

function in the Magic Sequence problem, you could do that only because the constraint `=`

has been reified. Reification allows us to create models with “soft constraints” or “conditional constraints”, i.e. one constraint has to be satisfed only if the second one is satisfied too, otherwise they both can be ignored. To do that, one has only to reify the constraints and connect them with the implication: `constraint1 → constraint2`

. Let's practice this quite useful technique :)

- Problem: There are two classes of objects (men and women, for example) that have to be matched according to their preferences. We say that a matching (marriage) is unstable if both spouses would prefer to be with somebody else. You can read more about this problem on wikipedia.
- Assignment:
- Download, extract, comprehend the model
- Add missing variables, constraints
- Give a high-five to your teacher :)