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In mathematics, constraint counting is counting the number of constraints in order to compare it with the number of variables, parameters, etc. that are free to be determined, the idea being that in most cases the number of independent choices that can be made is the excess of the latter over the former.
Global constraints [2] are constraints representing a specific relation on a number of variables, taken altogether. Some of them, such as the alldifferent constraint, can be rewritten as a conjunction of atomic constraints in a simpler language: the alldifferent constraint holds on n variables x 1 . . . x n {\displaystyle x_{1}...x_{n}} , and ...
A computationally difficult variation of 2-satisfiability, finding a truth assignment that maximizes the number of satisfied constraints, has an approximation algorithm whose optimality depends on the unique games conjecture, and another difficult variation, finding a satisfying assignment minimizing the number of true variables, is an ...
There are only 21853 pseudoprimes base 2 that are less than 2.5 × 10 10 (see page 1005 of [3]). This means that, for n up to 2.5 × 10 10, if 2 n −1 (modulo n) equals 1, then n is prime, unless n is one of these 21853 pseudoprimes. Some composite numbers (Carmichael numbers) have the property that a n − 1 is 1 (modulo n) for every a that ...
Without these constraints, one dual variable may take the value corresponding to the tuple =, = while another dual variable takes the value corresponding to =, =, which assigns a different value to . More generally, the constraints of the dual problem enforce the same values for all variables shared by two constraints.
In constraint satisfaction, constraint inference is a relationship between constraints and their consequences. A set of constraints D {\displaystyle D} entails a constraint C {\displaystyle C} if every solution to D {\displaystyle D} is also a solution to C {\displaystyle C} .
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 a higher value.
Phase-constrained least squares: all elements of must be real numbers, or multiplied by the same complex number of unit modulus. If the constraint only applies to some of the variables, the mixed problem may be solved using separable least squares [4] by letting = [] and = [] represent the unconstrained (1) and constrained (2) components.