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)) \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')|

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 $1 + \delta$ . Begin with the approximation ratio of A, $\frac{c_A(y)}{OPT_A(x)}$ . 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)}{OPT_A(x)} \le \frac{OPT_A(x) + \beta(c_B(y') - OPT_B(x'))}{OPT_A(x)}

Simplifying, and substituting the first condition, we have

\frac{c_A(y)}{OPT_A(x)} \le 1 + \alpha \beta \left( \frac{c_B(y')-OPT_B(x')}{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, $\frac{c_A(y)}{OPT_A(x)}$ . 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)}{OPT_A(x)} \ge \frac{OPT_A(x) - \beta(c_B(y') - OPT_B(x'))}{OPT_A(x)}

Simplifying, and substituting the first condition, we have

\frac{c_A(y)}{OPT_A(x)} \ge 1 - \alpha \beta \left( \frac{c_B(y')-OPT_B(x')}{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 Optimization Problems. Royal Institute of Technology, Sweden. ISBN 91-7170-082-X.
↑ Christos H. Papadimitriou; Mihalis Yannakakis (1988). "Optimization, Approximation, and Complexity Classes". STOC '88: Proceedings of the twentieth annual ACM Symposium on Theory of Computing. doi:10.1145/62212.62233.
1 2 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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