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expanded definition, said that storing inconsistent partial evaluation as is is not useful |
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Backtracking algorithms work by choosing an unassinged variable and recursively solve the problems obtained by assigning a value to this variable. Whenever the current partial solution is found inconsistent, the algorithm goes back to the previously assigned variable, as expected by recursion. A constraint learning algorithm differs because it tries to record some information, before backtracking, in form of a new constraint. This can reduce the further search because the subsequent search may encounter another partial solution that is inconsistent with this new constraint. If the algorithm has learned the new constraint, it will backtrack from this solution, while the original backtracking algorithm would do a subsequent search.
If the partial solution <math>x_1=a_1,\ldots,x_k=a_k</math> is inconsistent, the problem instance implies the constraint stating that <math>x_i=a_i</math> cannot be be true for all <math>i \in [1,k]</math> at the same time. However, recording this constraint is not useful, as this partial solution will not be encountered again due to the way backtracking proceed
On the other hand, if a subset of this evaluation is inconsistent, the corresponding constraint may be useful in the subsequent search, as the same subset of the partial evaluation may occur again in the search. For example, the algorithm may enounter an evaluation extending the subset <math>x_2=a_2, x_5=a_5, x_{k-1}=a_{k-1}</math> of the previous partial evaluation. If this subset is inconsistent and the algorithm has stored this fact in form of a constraint, no further search is needed to conclude that the new partial evaluation cannot be extended to form a solution.
The efficiency of constraint learning algorithm is balanced between two factors. On one hand, the more often a recorded constraint is violated, the more often block backtracking from doing useless search. As a result, algorithms search for small inconsistent subset of the current partial solution, as they correspond to constraints that are easier to violate. On the other hand, finding a small inconsistent subset of the current partial evaluation may require time, and the benefit may not be balanced by the subsequent reduction of the search time.
==See also==
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