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 {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)

|\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 an 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 $1+\delta$ . 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)}}\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

{\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 $\delta$ . 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)}}\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

{\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 $\delta '$ . 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.[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

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[Kann92-1] Kann, Viggo (1992). On the Approximability of NP-complete \mathrm{OPT}imization Problems. Royal Institute of Technology, Sweden. ISBN 978-91-7170-082-7.

[Papadimitriou88-2] 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-3] 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–.