L-reduction

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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
[math]\displaystyle{ \mathrm{OPT_B}(f(x)) \le \alpha \mathrm{OPT_A}(x) }[/math],
  • there exists a positive constant β such that for every solution y' to f(x)
[math]\displaystyle{ |\mathrm{OPT_A}(x)-c_A(g(y'))| \le \beta |\mathrm{OPT_B}(f(x))-c_B(y')| }[/math].

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 [math]\displaystyle{ 1 + \delta }[/math]. Begin with the approximation ratio of A, [math]\displaystyle{ \frac{c_A(y)}{\mathrm{OPT}_A(x)} }[/math]. 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

[math]\displaystyle{ \frac{c_A(y)}{\mathrm{OPT}_A(x)} \le \frac{\mathrm{OPT}_A(x) + \beta(c_B(y') - \mathrm{OPT}_B(x'))}{\mathrm{OPT}_A(x)} }[/math]

Simplifying, and substituting the first condition, we have

[math]\displaystyle{ \frac{c_A(y)}{\mathrm{OPT}_A(x)} \le 1 + \alpha \beta \left( \frac{c_B(y')-\mathrm{OPT}_B(x')}{\mathrm{OPT}_B(x')} \right) }[/math]

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

This meets the conditions for AP-reduction.

Proof (maximization case)

Let the approximation ratio of B be [math]\displaystyle{ \frac{1}{1 - \delta'} }[/math]. Begin with the approximation ratio of A, [math]\displaystyle{ \frac{c_A(y)}{\mathrm{OPT}_A(x)} }[/math]. 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

[math]\displaystyle{ \frac{c_A(y)}{\mathrm{OPT}_A(x)} \ge \frac{\mathrm{OPT}_A(x) - \beta(c_B(y') - \mathrm{OPT}_B(x'))}{\mathrm{OPT}_A(x)} }[/math]

Simplifying, and substituting the first condition, we have

[math]\displaystyle{ \frac{c_A(y)}{\mathrm{OPT}_A(x)} \ge 1 - \alpha \beta \left( \frac{c_B(y')-\mathrm{OPT}_B(x')}{\mathrm{OPT}_B(x')} \right) }[/math]

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

If [math]\displaystyle{ \frac{1}{1 - \alpha\beta\delta'} = 1+\epsilon }[/math], then [math]\displaystyle{ \frac{1}{1 - \delta'} = 1 + \frac{\epsilon}{\alpha\beta(1+\epsilon) - \epsilon} }[/math], 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

See also

References

  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. 
  2. ↑ Christos H. Papadimitriou; Mihalis Yannakakis (1988). "\mathrm{OPT}imization, Approximation, and Complexity Classes". doi:10.1145/62212.62233. 
  3. ↑ 3.0 3.1 Crescenzi, Pierluigi (1997). "A short guide to approximation preserving reductions". Proceedings of Computational Complexity. Twelfth Annual IEEE Conference. Washington, D.C.: IEEE Computer Society. pp. 262–. doi:10.1109/CCC.1997.612321. ISBN 9780818679070. http://dl.acm.org/citation.cfm?id=792302. 
  • 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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