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.

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:
 * 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
 * $$\mathrm{OPT_B}(f(x)) \le \alpha \mathrm{OPT_A}(x)$$,


 * there exists a positive constant β such that for every solution y'  to f(x)
 * $$|\mathrm{OPT_A}(x)-c_A(g(y'))| \le \beta |\mathrm{OPT_B}(f(x))-c_B(y')|$$.

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 $$1 + \delta$$. Begin with the approximation ratio of A, $$\frac{c_A(y)}{\mathrm{OPT}_A(x)}$$. 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
 * $$\frac{c_A(y)}{\mathrm{OPT}_A(x)} \le \frac{\mathrm{OPT}_A(x) + \beta(c_B(y') - \mathrm{OPT}_B(x'))}{\mathrm{OPT}_A(x)}$$

Simplifying, and substituting the first condition, we have
 * $$\frac{c_A(y)}{\mathrm{OPT}_A(x)} \le 1 + \alpha \beta \left( \frac{c_B(y')-\mathrm{OPT}_B(x')}{\mathrm{OPT}_B(x')} \right)$$

But the term in parentheses on the right-hand side actually equals $$\delta$$. Thus, the approximation ratio of A is $$1 + \alpha\beta\delta$$.

This meets the conditions for AP-reduction.

Proof (maximization case)
Let the approximation ratio of B be $$\frac{1}{1 - \delta'}$$. Begin with the approximation ratio of A, $$\frac{c_A(y)}{\mathrm{OPT}_A(x)}$$. We can remove absolute values around the third condition of the L-reduction definition since we know A and B are maximization problems. Substitute that condition to obtain
 * $$\frac{c_A(y)}{\mathrm{OPT}_A(x)} \ge \frac{\mathrm{OPT}_A(x) - \beta(c_B(y') - \mathrm{OPT}_B(x'))}{\mathrm{OPT}_A(x)}$$

Simplifying, and substituting the first condition, we have
 * $$\frac{c_A(y)}{\mathrm{OPT}_A(x)} \ge 1 - \alpha \beta \left( \frac{c_B(y')-\mathrm{OPT}_B(x')}{\mathrm{OPT}_B(x')} \right)$$

But the term in parentheses on the right-hand side actually equals $$\delta'$$. Thus, the approximation ratio of A is $$\frac{1}{1 - \alpha\beta\delta'}$$.

If $$\frac{1}{1 - \alpha\beta\delta'} = 1+\epsilon$$, then $$\frac{1}{1 - \delta'} = 1 + \frac{\epsilon}{\alpha\beta(1+\epsilon) - \epsilon}$$, 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

 * Dominating set: an example with α = β = 1
 * Token reconfiguration: an example with α = 1/5, β = 2