Approximation-preserving reduction
In computability theory and computational complexity theory, especially the study of approximation algorithms, an approximation-preserving reduction is an algorithm for transforming one optimization problem into another problem, such that the distance of solutions from optimal is preserved to some degree. Approximation-preserving reductions are a subset of more general reductions in complexity theory; the difference is that approximation-preserving reductions usually make statements on approximation problems or optimization problems, as opposed to decision problems.
Intuitively, problem A is reducible to problem B via an approximation-preserving reduction if, given an instance of problem A and a (possibly approximate) solver for problem B, one can convert the instance of problem A into an instance of problem B, apply the solver for problem B, and recover a solution for problem A that also has some guarantee of approximation.
Unlike reductions on decision problems, an approximation-preserving reduction must preserve more than the truth of the problem instances when reducing from one problem to another. It must also maintain some guarantee on the relationship between the cost of the solution to the cost of the optimum in both problems. To formalize:
Let and be optimization problems.
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