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

$\mathrm{O}\mathrm{P}{\mathrm{T}}_{\mathrm{B}}(f(x))\le \alpha \mathrm{O}\mathrm{P}{\mathrm{T}}_{\mathrm{A}}(x)$,

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

$|\mathrm{O}\mathrm{P}{\mathrm{T}}_{\mathrm{A}}(x)-c_A(g(y^{\prime }))|\le \beta |\mathrm{O}\mathrm{P}{\mathrm{T}}_{\mathrm{B}}(f(x))-c_B(y^{\prime })|$.

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 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, $\frac{c_A(y)}{{\mathrm{O}\mathrm{P}\mathrm{T}}_A(x)}$. 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

$\frac{c_A(y)}{{\mathrm{O}\mathrm{P}\mathrm{T}}_A(x)}\le \frac{{\mathrm{O}\mathrm{P}\mathrm{T}}_A(x)+\beta (c_B(y^{\prime })-{\mathrm{O}\mathrm{P}\mathrm{T}}_B(x^{\prime }))}{{\mathrm{O}\mathrm{P}\mathrm{T}}_A(x)}$

Simplifying, and substituting the first condition, we have

$\frac{c_A(y)}{{\mathrm{O}\mathrm{P}\mathrm{T}}_A(x)}\le 1+\alpha \beta \left(\frac{c_B(y^{\prime })-{\mathrm{O}\mathrm{P}\mathrm{T}}_B(x^{\prime })}{{\mathrm{O}\mathrm{P}\mathrm{T}}_B(x^{\prime })}\right)$

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 $\frac{1}{1-{\delta }^{\prime }}$. Begin with the approximation ratio of A, $\frac{c_A(y)}{{\mathrm{O}\mathrm{P}\mathrm{T}}_A(x)}$. 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

$\frac{c_A(y)}{{\mathrm{O}\mathrm{P}\mathrm{T}}_A(x)}\ge \frac{{\mathrm{O}\mathrm{P}\mathrm{T}}_A(x)-\beta (c_B(y^{\prime })-{\mathrm{O}\mathrm{P}\mathrm{T}}_B(x^{\prime }))}{{\mathrm{O}\mathrm{P}\mathrm{T}}_A(x)}$

Simplifying, and substituting the first condition, we have

$\frac{c_A(y)}{{\mathrm{O}\mathrm{P}\mathrm{T}}_A(x)}\ge 1-\alpha \beta \left(\frac{c_B(y^{\prime })-{\mathrm{O}\mathrm{P}\mathrm{T}}_B(x^{\prime })}{{\mathrm{O}\mathrm{P}\mathrm{T}}_B(x^{\prime })}\right)$

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

If $\frac{1}{1-\alpha \beta {\delta }^{\prime }}=1+ϵ$, then $\frac{1}{1-{\delta }^{\prime }}=1+\frac{ϵ}{\alpha \beta (1+ϵ)-ϵ}$, 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

See also

• MAXSNP
• Approximation-preserving reduction
• PTAS reduction

References

• 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