L-reduction


In computer science, particularly the study of approximation algorithms, an
L-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.

Definition

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:

Implication of PTAS reduction

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.

Proof (minimization case)

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.

Proof (maximization case)

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 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. Thus, the approximation ratio of A is.
If, then, which meets the requirements for PTAS reduction but not AP-reduction.

Other properties

L-reductions also imply P-reduction. 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.

Examples