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bucket elimination |
bucket elimination: removal of constraints |
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One way for evaluating this upper bound for a partial solution is to consider each soft constraint separately. For each soft constraint, the maximal possible value for any assignment to the unassigned variables is assumed. The sum of these values is an upper bound because the soft constraints cannot assume an higer value. It is exact because the maximal values of soft constraints may derive from different evaluations: a soft constraint may be maximal for <math>x=a</math> while another constraint is maximal for <math>x=b</math>.
===Russian
This method runs a branch-and-bound algorithm on <math>n</math> problems, where <math>n</math> is the number of variables. Each such problem is the subproblem obtained by dropping a sequence of variables <math>x_1,\ldots,x_i</math> from the original problem, along with the constraints containing them. After the problem on variables <math>x_{i+1},\ldots,x_n</math> is solved, its optimal cost can be used as an upper bound while solving the other problems,
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<!-- not exactly the correct notation, but clear enough -->
:<math>C(y_1=a_1,\ldots,y_n=a_n) = \max_{a} \sum_i C_i(x=a,y_1=a_1,\ldots,y_n=a_n)</math>
Bucket elimination works with an (arbitrary) ordering of the variables. Every variable is associated a bucket of constraints; the bucket of a variable contains all constraints having the variable has the highest in the order. Bucket elimination proceed from the last variable to the first. For each variable, all constraints of the bucket are replaced as above to remove the variable. The resulting constraint is then placed in the appropriate bucket.
==Reference==
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