Abstract
We prove that all NP problems over the reals with addition and order can be solved in polynomial time with the help of a boolean NP oracle. As a consequence, the “P = NP?” question over the reals with addition and order is equivalent to the classical question. For the reals with addition and equality only, the situation is quite different since P is known to be different from NP. Nevertheless, we prove similar transfer theorems for the polynomial hierarchy.
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References
L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation. Springer-Verlag, 1998.
F. Cucker and P. Koiran. Computing over the reals with addition and order: Higher complexity classes. Journal of Complexity, 11:358–376, 1995.
H. Fournier and P. Koiran. Are lower bounds easier over the reals? In Proc. 30th ACM Symposium on Theory of Computing, pages 507–513, 1998.
H. Fournier and P. Koiran. Lower bounds are not easier over the reals: Inside PH. LIP Research Report 99-21, Ecole Normale Supérieure de Lyon, 1999.
P. Koiran. Computing over the reals with addition and order. Theoretical Computer Science, 133(1):35–48, 1994.
K. Meer. A note on a P≠NP result for a restricted class of real machines. Journal of Complexity, 8:451–453, 1992.
F. Meyer auf der Heide. A polynomial linear search algorithm for the n-dimensional knapsack problem. Journal of the ACM, 31(3):668–676, 1984.
F. Meyer auf der Heide. Fast algorithms for n-dimensional restrictions of hard problems. Journal of the ACM, 35(3):740–747, 1988.
B. Poizat. Les Petits Cailloux. Nur Al-Mantiq Wal-Ma’rifah 3. Aléas, Lyon, 1995.
A. Schrijver. Theory of Linear and Integer Programming. Wiley, New-York, 1986.
M. Shub and S. Smale. On the intractability of Hilbert’s Nullstellensatz and an algebraic version of “P=NP”. Duke Mathematical Journal, 81(1):47–54, 1996.
S. Smale. On the P=NP problem over the complex numbers. Lecture given at the MSRI workshop on Complexity of Continuous and Algebraic Mathematics, November 1998. Lecture on video at http://www.msri.org.
L. G. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on Theory of Computing, pages 249–261, 1979.
L. G. Valiant. Reducibility by algebraic projections. In Logic and Algorithmic (an International Symposium held in honour of Ernst Specker), pages 365–380. Monographie n o 30 de L’Enseignement Mathématique, 1982.
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Fournier, H., Koiran, P. (2000). Lower Bounds Are Not Easier over the Reals: Inside PH. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_70
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DOI: https://doi.org/10.1007/3-540-45022-X_70
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