# 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
${\displaystyle \mathrm {OPT_{B}} (f(x))\leq \alpha \mathrm {OPT_{A}} (x)}$,
• there exists a positive constant β such that for every solution y' to f(x)
${\displaystyle |\mathrm {OPT_{A}} (x)-c_{A}(g(y'))|\leq \beta |\mathrm {OPT_{B}} (f(x))-c_{B}(y')|}$.

## Properties

### 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 a L-reduction from A to B, then A also has a PTAS.[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.[3]

#### Proof (minimization case)

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

${\displaystyle {\frac {c_{A}(y)}{\mathrm {OPT} _{A}(x)}}\leq {\frac {\mathrm {OPT} _{A}(x)+\beta (c_{B}(y')-\mathrm {OPT} _{B}(x'))}{\mathrm {OPT} _{A}(x)}}}$

Simplifying, and substituting the first condition, we have

${\displaystyle {\frac {c_{A}(y)}{\mathrm {OPT} _{A}(x)}}\leq 1+\alpha \beta \left({\frac {c_{B}(y')-\mathrm {OPT} _{B}(x')}{\mathrm {OPT} _{B}(x')}}\right)}$

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

This meets the conditions for AP-reduction.

#### Proof (maximization case)

Let the approximation ratio of B be ${\displaystyle {\frac {1}{1-\delta '}}}$. 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 maximization problems. Substitute that condition to obtain

${\displaystyle {\frac {c_{A}(y)}{\mathrm {OPT} _{A}(x)}}\geq {\frac {\mathrm {OPT} _{A}(x)-\beta (c_{B}(y')-\mathrm {OPT} _{B}(x'))}{\mathrm {OPT} _{A}(x)}}}$

Simplifying, and substituting the first condition, we have

${\displaystyle {\frac {c_{A}(y)}{\mathrm {OPT} _{A}(x)}}\geq 1-\alpha \beta \left({\frac {c_{B}(y')-\mathrm {OPT} _{B}(x')}{\mathrm {OPT} _{B}(x')}}\right)}$

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

If ${\displaystyle {\frac {1}{1-\alpha \beta \delta '}}=1+\epsilon }$, then ${\displaystyle {\frac {1}{1-\delta '}}=1+{\frac {\epsilon }{\alpha \beta (1+\epsilon )-\epsilon }}}$, which meets the requirements for PTAS reduction but not AP-reduction.

### Other properties

L-reductions also imply P-reduction.[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.