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On approximation scheme preserving reductibility and its applications

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  • Algorithms
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Foundation of Software Technology and Theoretical Computer Science (FSTTCS 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 880))

Abstract

In this paper we generalize the notion of polynomial-time approximation scheme preserving reducibility, called PTAS-reducibility, introduced in [4]. As a first application of this generalization, we prove the APX-completeness of a polynomially bounded optimization problem, that is, an APX problem whose measure function is bounded by a polynomial in the length of the instance and such that any APX problem is reducible to it. As far as we know, no such problem was known before. This result has been recently used in [10] to show that several natural optimization problem are APX-complete, such as max cut, max sat, min node cover, and min δ-TSP.

Successively, we apply the notion of APX-completeness to the study of the relative complexity of evaluating an ε-approximate value and computing an ε-approximate solution for any ε. We first show that if P ≠ NP ∩ coNP then the former question can be easier than the latter even if the optimization problem is NP-hard. We therefore give strong evidence that if an optimization problem is APX-complete then the two questions are both hard.

Research partially supported by the MURST project Algoritmi, Modelli di Calcolo, Strutture Informative.

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References

  1. Arora S., Lund C., Motwani R., Sudan M., Szegedy M.: Proof verification and hardness of approximation problems. Proc. 33-rd Ann. IEEE Symp. on Foundations of Computer Science (1992) 14–23

    Google Scholar 

  2. Ausiello G., Crescenzi P., Protasi M.: Approximate Solution of NP Optimization Problems Technical Report SI/RR-94/03, Universita di Roma “La Sapienza” (1994)

    Google Scholar 

  3. Crescenzi P., Kann V., Trevisan L.; On the structure of the classes NPO and APX. Manuscript in preparation (1994)

    Google Scholar 

  4. Crescenzi P., Panconesi A.: Completeness in approximation classes. Information and Computation 93 (1991) 241–262

    Google Scholar 

  5. Crescenzi P., Silvestri R.: Relative complexity of evaluating the optimum cost and constructing the optimum for maximization problems. Information Processing Letters 33 (1990) 221–226

    Google Scholar 

  6. Downey R.G., Fellows M.R.: Fixed parameter intractability. Proc. 7-th Structure in Complexity Theory (1992) 36–49

    Google Scholar 

  7. Hemaspaandra, L.A.: Complexity theory column 5: the not-ready-for-prime-time conjectures. SIGACT News (1994)

    Google Scholar 

  8. Johnson D.S.: Approximation algorithms for combinatorial problems. J. Comput. and Syst. Sci. 9 (1974) 256–278

    Google Scholar 

  9. Kann V.: On the approximability of NP-complete optimization problems. PhD thesis, Department of Numerical Analysis and Computing Science, Royal Institute of technology, Stockholm (1992)

    Google Scholar 

  10. Khanna S., Motwani R., Sudan M., Vazirani U.: On syntactic versus computational views of approximability. Proc. 35-th Ann. IEEE Symp. on Foundations of Computer Science (1994) to appear

    Google Scholar 

  11. Krentel M.W.: The complexity of optimization problems. J. Comput. and Syst. Sci. 36 (1988) 490–509

    Google Scholar 

  12. Orponen P., Mannila H.: On approximation preserving reductions: complete problems and robust measures. Technical Report C-1987-28, Department of Computer Science, University of Helsinki (1987)

    Google Scholar 

  13. Papadimitriou C.H., Yannakakis M.: Optimization, approximation, and complexity classes. J. Comput. and Syst. Sci. 43 (1991), 425–440

    Google Scholar 

  14. Paz A., Moran S.: Non deterministic polynomial optimization problems and their approximations. Theoret. Comput. Sci. 15 (1981) 251–277

    Google Scholar 

  15. Valiant L.: Relative complexity of checking and evaluating. Information Processing Letters 5 (1976) 20–23

    Google Scholar 

  16. Yannakakis M.: On the approximation of maximum satisfiability. Proc. 3-rd ACM-SIAM Symp. on Discrete Algorithms (1992) 1–9

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Scienze dell'Informazione, Università degli Studi di Roma, “La Sapienza”, Via Salaria 113, 00198, Roma, Italy

    Pierluigi Crescenzi & Luca Trevisan

Authors
  1. Pierluigi Crescenzi
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  2. Luca Trevisan
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P. S. Thiagarajan

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© 1994 Springer-Verlag Berlin Heidelberg

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Crescenzi, P., Trevisan, L. (1994). On approximation scheme preserving reductibility and its applications. In: Thiagarajan, P.S. (eds) Foundation of Software Technology and Theoretical Computer Science. FSTTCS 1994. Lecture Notes in Computer Science, vol 880. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58715-2_135

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  • DOI: https://doi.org/10.1007/3-540-58715-2_135

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58715-6

  • Online ISBN: 978-3-540-49054-8

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