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Exponential lower bounds for some NP-complete problems in a restricted linear decision tree model

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Abstract

LetV be a set inR n consisting of finitely many hyperplanes. The linear recognition problem given byV is to determine, using ternary comparisons of the form “f(x):0” wheref:R nR is a linear function, whether a pointxεR n is inV. We consider lower bounds on the number of comparisons whenV corresponds to some NP-complete problems. A technique is proposed for proving such bounds. If the tests “f(x):0” are restricted so thatf always defines some hyperplane inV, then some NP-complete problems are shown to have exponential lower bounds inn. Examples of larger classes of linear test functions are found such that the exponential lower bounds are still valid.

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References

  1. D. Dobkin and R. Lipton,A lower bound of 1/2 n 2 on linear search programs for the knapsack problem, J. Comput. System. Sci. 16 (1978), 413–417

    Google Scholar 

  2. D. P. Dobkin and R.J. Lipton,On the complexity of computations under varying sets of primitives. J. Comput. System Sci. 18 (1979), 86–91.

    Google Scholar 

  3. M. R. Garey and D. S. Johnson,Computers and Intractability. Freeman, San Francisco, 1979.

    Google Scholar 

  4. R. M. Karp,Reducibility among combinatorial problems. In:Complexity of Computer Computations (R. E. Miller and J. W. Thatcher, eds.), Plenum Press, New York-London, 1972, pp. 85–103.

    Google Scholar 

  5. M. O. Rabin,Proving simultaneous positivity of linear forms. J. Comput. System Sci. 6 (1972), 639–650.

    Google Scholar 

  6. E. M. Reingold,Computing the maxima and the median. Proc. 12th Ann. Symp. on Switching and Automata Theory (1971), 216–218.

  7. M. Snir,Proving lower bounds for linear decision trees. In:Automata, Languages, and Programming, Eighth Colloquium, Acre, July 1981. Springer Lecture Notes in Computer Science 115 (1981), 305–315.

    Google Scholar 

  8. A. C. Yao,On the parallel computation for the knapsack problem. Proc. 13th Ann. ACM Symp. on Theory of Computing (1981), 123–127.

  9. A. C. Yao and R. L. Rivest,On the polyhedral decision problem. SIAM J. on Computing 9 (1980), 343–347.

    Google Scholar 

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This work was supported by the Academy of Finland and by the Finnish Cultural Foundation. Partial support was provided by the National Science Foundation Grant MCS 79-15763 (Univ. of California).

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Ukkonen, E. Exponential lower bounds for some NP-complete problems in a restricted linear decision tree model. BIT 23, 181–192 (1983). https://doi.org/10.1007/BF02218439

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  • DOI: https://doi.org/10.1007/BF02218439

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