en>D.Lazard: /* Properties */ not easy to prove

2013-12-31T12:28:18Z

Properties: not easy to prove

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			'''L-reduction''' ("''linear reduction''") is a transformation of [[optimization problem]]s which linearly preserves approximability features. L-reductions in studies of approximability of [[optimization problem]]s play a similar role to that of [[Polynomial-time reduction\|polynomial reductions]] in the studies of [[computational complexity]] of [[decision problem]]s.

			The term ''L reduction'' is sometimes used to refer to [[log-space reduction]]s, by analogy with the complexity class [[L (complexity)\|L]], but this is a different concept.

			==Definition==
			Let A and B be [[optimization problem]]s and c<sub>A</sub> and c<sub>B</sub> 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>\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>\|\mathrm{OPT_A}(x)-c_A(g(y))\| \le \beta \|\mathrm{OPT_B}(f(x))-c_B(y)\|</math>.

			==Properties==
			Let a (1±ε)-[[approximation algorithm]] ''f'' for a problem A be such that <math>c_\mathrm{A}(f(x))</math> is at most <math>\varepsilon \cdot \mathrm{OPT_A}(x)</math> away from <math>\mathrm{OPT_A}(x)</math>, for every instance ''x.'' (In this notation, + implicitly means a minimization problem, and <math>-</math> means a maximization problem.)

			The main point of an L-reduction is the following: given a (''f'',''g'',α,β) L-reduction from problem A to problem B, and a (1±ε)-[[approximation algorithm]] for B, we obtain a polynomial-time (1±δ)-[[approximation algorithm]] for A where <math>\delta = \alpha \beta \varepsilon</math>.<ref name=Kann92>{{cite book
			\| last1 = Kann \| first1 = Viggo
			\| year = 1992
			\| title = On the Approximability of NP-complete Optimization Problems
			\| publisher = Royal Institute of Technology, Sweden
			\| isbn = 91-7170-082-X
			}}</ref><ref name=Papadimitriou88>{{cite conference
			\| author= Christos H. Papadimitriou
			\| coauthors = Mihalis Yannakakis
			\| booktitle = STOC '88: Proceedings of the twentieth annual ACM Symposium on Theory of Computing
			\| title = Optimization, Approximation, and Complexity Classes
			\| year = 1988
			\| doi = 10.1145/62212.62233
			}}</ref>
			This implies that if B has a [[polynomial-time approximation scheme]] then so does A.

			==See also==
			* [[MAXSNP]]
			* [[PTAS reduction]]
			* [[Dominating set#L-reductions]]: an example with α = β = 1

			==References==
			{{reflist}}
			* Pierluigi Crescenzi: A Short Guide to Approximation Preserving Reductions. In: Proceedings of the Twelfth Annual IEEE Conference on Computational Complexity, June 24-27, 1997, Ulm, Germany. Pages 262-273. IEEE Computer Society Press, 1997. ISBN 0-8186-7907-7
			* 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

			{{comp-sci-theory-stub}}

			[[Category:Computational complexity theory]]
			[[Category:Approximation algorithms]]

en>Lockley: add ref tag

2012-07-28T07:14:45Z

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en>D.Lazard: /* Properties */ not easy to prove

en>Lockley: add ref tag