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Notes on polynomial levelability

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Abstract

We consider the polynomial levelability with respect to approximation algorithms (PLAA). A setA isPLAA if given any approximation algorithmα forA and a polynomialp, there are another approximation algorithmβ forA and a polynomialq such that for infinitely many inputsx,α acceptsx but has running time greater thanp(|x|) andβ acceptsx within timeq(|x|). In this paper, an algorithmα is called an approximation algorithm forA if the symmetric differenceA Δ L(α) is sparse, whereL(α) is the set of strings recognized byα. We prove that all naturalNP-complete sets arePLAA unlessP=NP and allEXP-complete sets arePLAA.

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References

  1. L. Berman, On the Structure of Complete Sets: Almost-everywhere Complexity and Infinitely often Speedup, 17thIEEE Sym. Foundation of Computer Science, 1976, 76–80.

  2. L. Berman and J. Hartmanis, On Isomorphisms and Density of NP and Other Complete Sets,SIAM Journal on Computing,6 (1977), 305–327.

    Google Scholar 

  3. R. Book and D. -Z. Du, The Existence and Density of Generalized Complexity Cores,Journal of ACM,34: 3 (1987), 718–730.

    Google Scholar 

  4. S. Fortune, A Note on Sparse Complete Sets,SIAM Journal on Computing,8 (1979), 431–433.

    Google Scholar 

  5. D. Joseph and P. Young, Some Remarks on Witness Functions for Non-polynomial and Non-complete Sets in NP,Theoretical Computer Science,39 (1985), 225–237.

    Google Scholar 

  6. K. Ko, Non-levelable Sets and Immune Sets in the Accepting Density Hierarchy in NP,Math. Syst. Theory,18 (1985), 189–205.

    Google Scholar 

  7. S. Mahaney, Sparse Complete Sets for NP: Solution of a Conjecture of Berman and Hartmanis,Journal of Comput. and Syst. Sci.,25 (1982), 130–143.

    Google Scholar 

  8. P. Orponen, D. Russo, and U. Schöning, Optimal Approximations and Polynomially Levelable Sets,SIAM Journal on Computing,15 (1986), 399–480.

    Google Scholar 

  9. U. Schöning, Complete Sets and Closeness to Complexity Classes. Submitted for Publication.

  10. O. Watanabe, On One-one P-equivalence Relations,Theoretical Computer Science,38 (1985), 157–165.

    Google Scholar 

  11. Y. Yesha, On Certain Polynomial-time Truth-table Reducibilities of Complete Sets to Sparse Sets,SIAM Journal on Computing,12 (1983), 411–425.

    Google Scholar 

  12. P. Young, Some Structural Properties of Polynomial Reducibilities and Sets in NP,Proc. of the 15th ACM Symp. on Theory of Computing, 1983, 392–401.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Academia Sinica, China

    Du Dingzhu

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  1. Du Dingzhu
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Additional information

This work was supported in part by the NSF under Grant DCR 83-12472 and was done while the author studied at the Department of Mathematics, University of California at Santa Barbara. The work was also supported by Chinese National Science Foundation

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Du, D. Notes on polynomial levelability. Acta Mathematicae Applicatae Sinica 4, 122–130 (1988). https://doi.org/10.1007/BF02006060

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  • DOI: https://doi.org/10.1007/BF02006060

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