Abstract
We first prove the existence of natural Poly-APX-complete problems, for both standard and differential approximation paradigms, under already defined and studied suitable approximation preserving reductions. Next, we devise new approximation preserving reductions, called FT and DFT, respectively, and prove that, under these reductions, natural problems are PTAS-complete, always for both standard and differential approximation paradigms. To our knowledge, no natural problem was known to be PTAS-complete and no problem was known to be Poly-APX-complete until now. We also deal with the existence of intermediate problems under FT- and DFT-reductions and we show that such problems exist provided that there exist NPO-intermediate problems under Turing-reduction. Finally, we show that min coloring is APX-complete for the differential approximation.
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References
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and approximation. In: Combinatorial optimization problems and their approximability properties, Springer, Berlin (1999)
Garey, M.R., Johnson, D.S.: Computers and intractability. In: A guide to the theory of NP-completeness, W. H. Freeman, San Francisco (1979)
Bazgan, C., Escoffier, B., Paschos, V.Th.:Completeness in standard and differential approximation classes: Poly-(D)APX- and (D)PTAS-completeness. Cahier du LAMSADE 217, LAMSADE, Université Paris-Dauphine, Available on (2004), http://www.lamsade.dauphine.fr/cahiers.html
Crescenzi, P., Panconesi, A.: Completeness in approximation classes. Information and Computation 93, 241–262 (1991)
Ausiello, G., Crescenzi, P., Protasi, M.: Approximate solutions of NP optimization problems. Theoret. Comput. Sci. 150, 1–55 (1995)
Crescenzi, P., Trevisan, L.: On approximation scheme preserving reducibility and its applications. Theory of Computing Systems 33, 1–16 (2000)
Ausiello, G., Bazgan, C., Demange, M., Paschos, V.Th.: Completeness in differential approximation classes. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 179–188. Springer, Heidelberg (2003)
Khanna, S., Motwani, R., Sudan, M., Vazirani, U.: On syntactic versus computational views of approximability. SIAM J 28, 164–191 (1998)
Ausiello, G., Bazgan, C., Demange, M., Paschos, V. Th.: Completeness in differential approximation classes. Cahier du LAMSADE 204, LAMSADE, Université Paris-Dauphine (2003), Available on, http://www.lamsade.dauphine.fr/cahiers.html
Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. Assoc. Comput. Mach. 41, 153–180 (1994)
Demange, M., Monnot, J., Paschos, V.Th.: Bridging gap between standard and differential polynomial approximation: the case of bin-packing. Appl. Math. Lett. 12, 127–133 (1999)
Orponen, P., Mannila, H.: On approximation preserving reductions: complete problems and robust measures. Technical Report C-1987-28, Dept. of Computer Science, University of Helsinki, Finland (1987)
Halldórsson, M.M.: Approximating discrete collections via local improvements. In: Proc. Symposium on Discrete Algorithms, SODA, pp. 160–169 (1995)
Feige, U., Kilian, J.: Zero knowledge and the chromatic number. In: Proc. Conference on Computational Complexity, pp. 278–287 (1996)
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Bazgan, C., Escoffier, B., Paschos, V.T. (2004). Poly-APX- and PTAS-Completeness in Standard and Differential Approximation. In: Fleischer, R., Trippen, G. (eds) Algorithms and Computation. ISAAC 2004. Lecture Notes in Computer Science, vol 3341. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30551-4_13
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DOI: https://doi.org/10.1007/978-3-540-30551-4_13
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