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While the width of the graph affects the time required for solving the subproblems in each node, the size of the separator affects the size of the constraints that are passed between nodes. Indeed, these constraints have the separators as scope. As a result, a constraint over a separator of size <math>n</math> may require size <math>d^n</math> to be stored, if all variables have ___domain of size <math>d</math>.
===Memory/time tradeoff===
The algorithm for solving a problem from a tree decomposition is effecively composed of two parts: first, constraints on the separators are created and passed between nodes; second, the subproblems at each node are solved. Different strategies can be used for creating the constraints required by the first part and to solve the individual subproblems in the second part. In particular, the creating the summary constraints can be done using variable elimination, which is a form of inference, while the subproblem can be solved by search (backtracking, etc.)
A problem with this algorithm is that the constraints passed between nodes can be of size exponential in the size of the separator (the set of variables the two nodes share). The memory required for storing these constraints can be decreased by using a tree decomposition with small separators. Such tree decompositions may however have width (number of nodes in each node) larger than optimal.
For a given tree decomposition, a fixed maximal allowed separator size can be enforced by joining all pairs of nodes whose separator is larger than this size. Merging two nodes usually produces a node with an associated set of variables larger than those of the two nodes. This may increase the width of the tree. However, this merging does not change the separators of the tree other than removing the separator between the two merged nodes.
The latter is a consequence of acyclicity: two joined nodes cannot be joined to the same other node. If <math>n_1</math> and <math>n_2</math> are two nodes to be merged and <math>N_1</math> and <math>N_2</math> are the sets of nodes joined to them, then <math>N_1 \cap N_2=\emptyset</math>, as otherwise there would be cycle in the tree. As a result, the node obtained by merging <math>n_1</math> and <math>n_2</math> will be joined to each of the nodes of <math>N_1 \cup N_2</math>. As a result, the separators of this merged node are exactly the separators of the two original nodes.
As a result, merging a pair of nodes joined by a separator does not change the other separators. As a result, a fixed maximal separator size can be enforced by first calculating all separator sizes and then iteratively merging any pair of nodes having a separator larger than a given amount, and the size of the separators do not need to be recalculated during execution.
===Separable components===
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