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: 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.

[1][2] 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.

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

We can remove absolute values around the third condition of the L-reduction definition since we know A and B are maximization 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

, which meets the requirements for PTAS reduction but not AP-reduction.

[3] One may deduce that L-reductions imply PTAS reductions from this fact and the fact that P-reductions imply PTAS reductions.

L-reductions preserve membership in APX for the minimizing case only, as a result of implying AP-reductions.

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