# 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 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:

- 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

- ,

- there exists a positive constant β such that for every solution
*y'*to*f*(*x*)

- .

## 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 . 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.[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.

## Examples

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

## References

- Kann, Viggo (1992).
*On the Approximability of NP-complete \mathrm{OPT}imization Problems*. Royal Institute of Technology, Sweden. ISBN 978-91-7170-082-7. - Christos H. Papadimitriou; Mihalis Yannakakis (1988). "\mathrm{OPT}imization, Approximation, and Complexity Classes".
*STOC '88: Proceedings of the twentieth annual ACM Symposium on Theory of Computing*. doi:10.1145/62212.62233. - Crescenzi, Pierluigi (1997). "A Short Guide To Approximation Preserving Reductions".
*Proceedings of the 12th Annual IEEE Conference on Computational Complexity*. Washington, D.C.: IEEE Computer Society: 262–.

- G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, M. Protasi. Complexity and Approximation. Combinatorial optimization problems and their approximability properties. 1999, Springer. ISBN 3-540-65431-3