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 c_A and c_B their respective cost functions. A pair of functions f and g is an L-reduction if all of the following conditions are met:

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 ${\displaystyle 1+\delta }$. Begin with the approximation ratio of A, ${\displaystyle {\frac {c_{A}(y)}{\mathrm {OPT} _{A}(x)}}}$. 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 ${\displaystyle \delta }$. Thus, the approximation ratio of A is ${\displaystyle 1+\alpha \beta \delta }$.

This meets the conditions for AP-reduction.