Probabilistic approximation of some NP optimization problems by finite-state machines

  • Dawei Hong
  • Jean-Camille Birget
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1269)


We introduce a subclass of NP optimization problems which contains, e.g., Bin Covering and Bin Packing. For each problem in this subclass we prove that with probability tending to 1 as the number of input items tends to infinity, the problem is approximable up to any given constant factor ε > 0 by a finite-state machine. More precisely, let II be a problem in our subclass of NP optimization problems, and let I be an input represented by a sequence of n independent identically distributed random variables with a fixed distribution. Then for any ε > 0 there exists a finite-state machine which does the following: On a random input I the finite-state machine produces a feasible solution whose objective value M(I) satisfies
$$P\left( {\frac{{|Opt(I) - M(I)|}}{{\max \{ Opt(I),M(I)\} }} \geqslant \varepsilon } \right) \leqslant K\exp ( - hn)$$
when n is large enough. Here K and h are positive constants.


NP- optimization problems approximation probabilistic algorithms finite-state machines 


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  1. 1.
    K. Azuma, Weighted sums of certain dependent variables, Tohoku Mathematical Journal 3 (1965) 357–367.Google Scholar
  2. 2.
    P. Crescenzi and A. Panconesi, Completeness in approximation classes, Information and Computation 93 (1991) 241–262.Google Scholar
  3. 3.
    E. G. Coffman, Jr., M. R. Garey, and D.S. Johnson, Approximation algorithms for bin packing — an updated survey. In G. Ausiello, M. Lucertini, and P. Serafini, editors,Algorithm Design for Computer System Design, CISM Courses and Lectures no. 284, pp. 49–106, Springer-Verlag 1984.Google Scholar
  4. 4.
    E. G. Coffman, Jr. and G.S. Lueker, Probabilistic Analysis of Packing and Partitioning Algorithms, John Wiley & Sons, 1991.Google Scholar
  5. 5.
    P. Gaenssler, Empirical Processes, Lecture Note — Monograph Series, Vol. 3, Institute of Mathematical Statistics, Hayward, CA, 1983.Google Scholar
  6. 6.
    S. Han, D. Hong and J. Y-T. Leung, Probabilistic analysis of a bin covering algorithm, Operations Research Letters Vol. 18 No. 4 (1995).Google Scholar
  7. 7.
    W. Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association, 58 (1965) 13–30.Google Scholar
  8. 8.
    D. Hong and J. Y-T. Leung, Probabilistic analysis of k-dimensional packing algorithms, Information Processing Letters 55 (1995) 17–24.Google Scholar
  9. 9.
    D. S. Johnson, Approximation algorithms for combinatorial problems, Journal of Computer and System Sciences 9 (1974) 256–278.Google Scholar
  10. 10.
    J. F. C. Kingman, Subadditive processes, Lecture Notes in Mathematics 539, pp. 168–222, Springer-Verlag, 1976.Google Scholar
  11. 11.
    C. H. Papadimitriou, Computational Complexity, Addison — Wesley, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Dawei Hong
    • 1
  • Jean-Camille Birget
    • 2
  1. 1.Dept. of Math. & Computer ScienceSouthwest State UniversityMarshallUSA
  2. 2.Dept. of Computer Science & Eng.University of NebraskaLincolnUSA

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