Constraint Programming: Basic Modelling Techniques
Symmetry Breaking
The problem is said to contain symmetry if there exist classes of equivalent solutions — solutions, which are called symmetrical because there exists 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 a much bigger search space. Symmetry breaking prunes the search space by removing symmetries from the problem.
All files required to solve the assignments are available via the repository, so clone it first.
Graph Coloring

Assignment:
Look at and comprehend lab3/graph_coloring/graph_coloring.mzn
model.
Try to solve the basic_data.dzn
instance.
You can use the model created during previous classes
There is a chance, that problem would too difficult to be solved in a reasonable time.
File data_with_clique.dzn
includes info about the largest clique in the graph
Improve model to make use of the info about the largest clique
Try to solve the problem again.
MultiKnapsack Problem
Redundant Constraints
There is a good chance the problem can be defined in more than 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 an impact on the number or quality of the solutions. The only reason to include them in 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, a redundant constraint may boost this process).
Magic Sequence

Assignment:
Look at and comprehend lab3/magic_series/magic_series.mzn
Add redundant constraints, hints:
Compare solving time with and without the redundant constraints.
Smile with satisfaction
Channeling
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.

Assignment:
Look at and comprehend lab3/n_queens/n_queens.mzn
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:
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
Compare running time again
Give yourself a
selffive,
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) ];
Reified Constraints
Reification in Constraint Programming means treating the constraint as a firstorder 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 :)
Stable Marriage Problem
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:
Look at and comprehend lab3/stablemarriage/stablemarriage.mzn
Add missing variables, constraints
Give a highfive to your teacher :)