Abstract
We prove completeness results for twenty-three problems in semilinear geometry. These results involve semilinear sets given by additive circuits as input data. If arbitrary real constants are allowed in the circuit, the completeness results are for the Blum-Shub-Smale additive model of computation. If, in contrast, the circuit is constant-free, then the completeness results are for the Turing model of computation. One such result, the P NP[log]-completeness of deciding Zariski irreducibility, exhibits for the first time a problem with a geometric nature complete in this class.
A full version of this paper can be obtained at http://www-math.upb.de/agpb
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Adian, S.I.: Unsolvability of certain algorithmic problems in the theory of groups. Trudy Moskov. Math. Obshch. 6, 231–298 (1957) (in Russian)
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)
Bürgisser, P., Cucker, F.: Counting complexity classes for numeric computations I: Semilinear sets. SIAM J. Comp. 33, 227–260 (2004)
Chandra, A., Stockmeyer, L., Vishkin, U.: Constant depth reducibility. SIAM J. Comp. 13, 423–439 (1984)
Cucker, F., Koiran, P.: Computing over the reals with addition and order: Higher complexity classes. Journal of Complexity 11, 358–376 (1995)
Fournier, H., Koiran, P.: Are lower bounds easier over the reals? In: Proc. 30th ACM STOC, pp. 507–513 (1998)
Hemaspaandra, E., Hemaspaandra, L.A., Rothe, J.: Exact Analysis of Dodgson Elections: Lewis Carroll’s 1876 Voting System is Complete for Parallel Access to NP. Journal of the ACM, 806–825 (1997)
Khachijan, L.G.: A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR 244, 1093–1096 (1979); in Russian, English translation in Soviet Math. Dokl., 20, 191–194 (1979)
Koiran, P.: Computing over the reals with addition and order. Theoretical Computer Science 133, 35–47 (1994)
Krentel, M.W.: The complexity of optimization problems. In: Proc. 18th ACM Symp. on the Theory of Computing, pp. 79–86 (1986)
Meyer auf der Heide, F.: A polynomial linear search algorithm for the n-dimensional knapsack problem. J. ACM 31, 668–676 (1984)
Papadimitriou, C.H.: On the complexity of unique solutions. J. ACM 31, 392–400 (1984)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)
Papadimitriou, C.H., Zachos, S.: Two remarks on the power of counting. In: Cremers, A.B., Kriegel, H.-P. (eds.) GI-TCS 1983. LNCS, vol. 145, pp. 269–276. Springer, Heidelberg (1982)
Rabin, M.: Recursive unsolvability of group theoretic problems. Ann. of Math. 67(2), 172–194 (1958)
Shafarevich, I.R.: Basic Algebraic Geometry. Springer, Heidelberg (1974)
Smale, S.: Mathematical problems for the next century. Mathematical Intelligencer 20, 7–15 (1998)
Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34, 250–256 (1986)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bürgisser, P., Cucker, F., de Naurois, P.J. (2005). The Complexity of Semilinear Problems in Succinct Representation. In: Liśkiewicz, M., Reischuk, R. (eds) Fundamentals of Computation Theory. FCT 2005. Lecture Notes in Computer Science, vol 3623. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537311_42
Download citation
DOI: https://doi.org/10.1007/11537311_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28193-1
Online ISBN: 978-3-540-31873-6
eBook Packages: Computer ScienceComputer Science (R0)