L-reduction

In computer science, particularly the study of approximation algorithms, an L-reduction ("linear reduction") is a transformation of optimization problems which linearly preserves approximability features; it is one type of approximation-preserving reduction. L-reductions in studies of approximability of optimization problems play a similar role to that of polynomial reductions in the studies of computational complexity of decision problems. The term L reduction is sometimes used to refer to log-space reductions, by analogy with the complexity class L, but this is a different concept. Let A and B be optimization problems and cA and cB their respective cost functions. A pair of functions f and g is an L-reduction if all of the following conditions are met: functions f and g are computable in polynomial time, if x is an instance of problem A, then f(x) is an instance of problem B, if y' is a solution to f(x), then g(y' ) is a solution to x, there exists a positive constant α such that there exists a positive constant β such that for every solution y' to f(x) An L-reduction from problem A to problem B implies an AP-reduction when A and B are minimization problems and a PTAS reduction when A and B are maximization problems. In both cases, when B has a PTAS and there is an L-reduction from A to B, then A also has a PTAS. This enables the use of L-reduction as a replacement for showing the existence of a PTAS-reduction; Crescenzi has suggested that the more natural formulation of L-reduction is actually more useful in many cases due to ease of usage. Let the approximation ratio of B be . Begin with the approximation ratio of A, . We can remove absolute values around the third condition of the L-reduction definition since we know A and B are minimization problems. Substitute that condition to obtain Simplifying, and substituting the first condition, we have But the term in parentheses on the right-hand side actually equals . Thus, the approximation ratio of A is . This meets the conditions for AP-reduction.