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Constraint Programming: Basic Modelling Tenchniques
Symmetry Breaking
The problem is said to contain symmetry if there are 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.
Graph Coloring
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Assignment:
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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
Improve model to make use of the info about the largest clique
Try to solve the problem again.
Multi-Knapsack Problem
Redundant Constraints
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).
Magic Sequence
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Assignment:
-
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 one 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.
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Assignment:
Download, extract and comprehend
the model (or use your own)
Add the other problem definition to the problem
Channel constraints from the both models:
Compare running time of the normal and channeled model
Give yourself a high five, however new solvers are good enough to solve n-queens without the channeling. This technique is still valid for the more complicated problems
Search Modeling
So far we haven't talked about the way solver looks for the solution. There are many different techniques to solve to constraint programming problem, however basic techniques often perform a DFS (backtracking) search with two steps at every node:
select variable — choose, which variable will receive a value in this step
select value — choose, which value from the variable's domain will be chosen
You may control this procedure in MiniZinc using search annotations just after the solve keyword. e.g.
solve :: int_search(array, first_fail, indomain_min, complete) satisfy;
mean that że integer (int
) variables from the array
should be search exhaustively (complete
) according to the simple strategy:
In order to define more interesting search strategies one has to use so called MiniSearch language, which still isn't a part of the MiniZincIDE package.
N-Queens Again