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A THEORY OF COMPLEXITY, CONDITION, AND ROUNDOFF

  • FELIPE CUCKER (a1)

Abstract

We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub and Smale for computations over $\mathbb{R}$ (which in turn followed those of the classical, discrete, complexity theory as laid down by Cook, Karp, and Levin, among others). In particular, we focus on complexity classes of decision problems and, paramount among them, on appropriate versions of the classes $\mathsf{P}$ , $\mathsf{NP}$ , and $\mathsf{EXP}$ of polynomial, nondeterministic polynomial, and exponential time, respectively. We prove some basic relationships between these complexity classes, and provide natural NP-complete problems.

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Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

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[1]Allender, E., Bürgisser, P., Kjeldgaard-Pedersen, J. and Miltersen, P. B., ‘On the complexity of numerical analysis’, SIAM J. Comput. 38(5) (2008/09), 19872006.
[2]Beltrán, C. and Pardo, L. M., ‘Smale’s 17th problem: average polynomial time to compute affine and projective solutions’, J. Amer. Math. Soc. 22(2) (2009), 363385.
[3]Beltrán, C. and Pardo, L. M., ‘Fast linear homotopy to find approximate zeros of polynomial systems’, Found. Comput. Math. 11(1) (2011), 95129.
[4]Blum, L., ‘Lectures on a theory of computation and complexity over the reals (or an arbitrary ring)’, in: Lectures in the Sciences of Complexity II (ed. Jen, E.) (Addison-Wesley, Redwood City, CA, 1990), 147.
[5]Blum, L., Cucker, F., Shub, M. and Smale, S., Complexity and Real Computation (Springer, New York, 1998).
[6]Blum, L., Shub, M. and Smale, S., ‘On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines’, Bull. Amer. Math. Soc. 21 (1989), 146.
[7]Braverman, M. and Cook, S., ‘Computing over the reals: foundations for scientific computing’, Notices Amer. Math. Soc. 53(3) (2006), 318329.
[8]Bürgisser, P. and Cucker, F., ‘Exotic quantifiers, complexity classes, and complete problems’, Found. Comput. Math. 9 (2009), 135170.
[9]Bürgisser, P. and Cucker, F., ‘On a problem posed by Steve Smale’, Ann. Math. 174 (2011), 17851836.
[10]Bürgisser, P. and Cucker, F., Condition, Grundlehren der mathematischen Wissenschaften, Vol. 349, (Springer, Berlin, 2013).
[11]Bürgisser, P., Cucker, F. and Lotz, M., ‘Coverage processes on spheres and condition numbers for linear programming’, Ann. Probab. 38 (2010), 570604.
[12]Cheung, D. and Cucker, F., ‘A new condition number for linear programming’, Math. Program. 91 (2001), 163174.
[13]Cheung, D., Cucker, F. and Ye, Y., ‘Linear programming and condition numbers under the real number computation model’, in: Handbook of Numerical Analysis, Vol. XI (eds. Ciarlet, Ph. and Cucker, F.) (North-Holland, Amsterdam, 2003), 141207.
[14]Cobham, A., ‘The intrinsic computational difficulty of problems’, in: International Congress for Logic, Methodology, and the Philosophy of Science (ed. Bar-Hillel, Y.) (North-Holland, Amsterdam, 1964), 2430.
[15]Cook, S., ‘The complexity of theorem proving procedures’, in: 3rd Annual ACM Symposium on the Theory of Computing (Assoc. Comput. Mach., New York, 1971), 151158.
[16]Cook, S., ‘The P versus NP problem’, in: The Millennium Prize Problems (Clay Math. Inst., Cambridge, MA, 2006), 87104.
[17]Cucker, F., ‘P̸ = NC’, J. Complexity 8 (1992), 230238.
[18]Cucker, F. and Koiran, P., ‘Computing over the reals with addition and order: higher complexity classes’, J. Complexity 11 (1995), 358376.
[19]Cucker, F., Krick, T., Malajovich, G. and Wschebor, M., ‘A numerical algorithm for zero counting. I. Complexity and accuracy’, J. Complexity 24 (2008), 582605.
[20]Cucker, F., Krick, T., Malajovich, G. and Wschebor, M., ‘A numerical algorithm for zero counting. II. Distance to ill-posedness and smoothed analysis’, J. Fixed Point Theory Appl. 6 (2009), 285294.
[21]Cucker, F. and Peña, J., ‘A primal-dual algorithm for solving polyhedral conic systems with a finite-precision machine’, SIAM J. Optim. 12 (2002), 522554.
[22]Cucker, F. and Smale, S., ‘Complexity estimates depending on condition and round-off error’, J. ACM 46 (1999), 113184.
[23]Cucker, F. and Torrecillas, A., ‘Two P-complete problems in the theory of the reals’, J. Complexity 8 (1992), 454466.
[24]Demmel, J., ‘On condition numbers and the distance to the nearest ill-posed problem’, Numer. Math. 51 (1987), 251289.
[25]Demmel, J. W., Applied Numerical Linear Algebra (SIAM, Philadelphia, PA, 1997).
[26]Eckart, C. and Young, G., ‘The approximation of one matrix by another of lower rank’, Psychometrika 1 (1936), 211218.
[27]Edelman, A., ‘Eigenvalues and condition numbers of random matrices’, SIAM J. Matrix Anal. Appl. 9 (1988), 543556.
[28]Edmonds, J., ‘Paths, trees, and flowers’, Canad. J. Math. 17 (1965), 449467.
[29]Goffin, J.-L., ‘The relaxation method for solving systems of linear inequalities’, Math. Oper. Res. 5 (1980), 388414.
[30]Goldreich, O., Computational Complexity (Cambridge University Press, Cambridge, 2008), A conceptual perspective.
[31]Hartmanis, J., Lewis, P. L. and Stearns, R. E., ‘Hierarchies of memory-limited computations’, in: 6th IEEE Symposium on Switching Circuit Theory and Logic Design (IEEE Comput. Soc., Long Beach, CA, 1965), 179190.
[32]Hartmanis, J. and Stearns, R. E., ‘On the computational complexity of algorithms’, Trans. Amer. Math. Soc. 117 (1965), 285306.
[33]Heintz, J., Roy, M.-F. and Solerno, P., ‘Sur la complexité du principe de Tarski–Seidenberg’, Bull. Soc. Math. France 118 (1990), 101126.
[34]Hestenes, M. R. and Stiefel, E., ‘Methods of conjugate gradients for solving linear systems’, J. Research Nat. Bur. Standards 49(1953) (1952), 409436.
[35]Higham, N., Accuracy and Stability of Numerical Algorithms (SIAM, Philadelphia, PA, 1996).
[36]Homer, S. and Selman, A. L., Computability and complexity theory, second edn, Texts in Computer Science, (Springer, New York, 2011).
[37]Karp, R. M., ‘Reducibility among combinatorial problems’, in: Complexity of Computer Computations (eds. Miller, R. and Thatcher, J.) (Plenum Press, 1972), 85103.
[38]Ko, K.-I., Complexity Theory of Real Functions, Progress in Theoretical Computer Science, (Birkhäuser Boston, Inc., Boston, MA, 1991).
[39]Koiran, P., ‘Computing over the reals with addition and order’, Theor. Comput. Sci. 133 (1994), 3547.
[40]Ladner, R. E., ‘The circuit value problem is log space complete for ℙ’, SIGACT News 7 (1975), 1820.
[41]Levin, L., ‘Universal sequential search problems’, Probl. Pered. Inform. IX 3 (1973), 265266. (in Russian), (English translation in Problems of Information Trans. 9, 3; corrected translation in [58]).
[42]Miller, W., ‘Computational complexity and numerical stability’, SIAM J. Comput. 4 (1975), 97107.
[43]Papadimitriou, C. H., Computational Complexity (Addison-Wesley, Redwood City, CA, 1994).
[44]Poizat, B., Les Petits Cailloux (Aléa, Paris, 1995).
[45]Renegar, J., ‘On the computational complexity and geometry of the first-order theory of the reals. Part I’, J. Symbolic Comput. 13 (1992), 255299.
[46]Renegar, J., ‘Is it possible to know a problem instance is ill-posed?’, J. Complexity 10 (1994), 156.
[47]Renegar, J., ‘Some perturbation theory for linear programming’, Math. Program. 65 (1994), 7391.
[48]Renegar, J., ‘Incorporating condition measures into the complexity theory of linear programming’, SIAM J. Optim. 5 (1995), 506524.
[49]Renegar, J., ‘Linear programming, complexity theory and elementary functional analysis’, Math. Program. 70 (1995), 279351.
[50]Shub, M. and Smale, S., ‘Complexity of Bézout’s Theorem I: geometric aspects’, J. Amer. Math. Soc. 6 (1993), 459501.
[51]Shub, M. and Smale, S., ‘Complexity of Bézout’s Theorem II: volumes and probabilities’, in: Computational Algebraic Geometry, (eds. Eyssette, F. and Galligo, A.) Progress in Mathematics, Vol. 109 (Birkhäuser, Basel, 1993), 267285.
[52]Shub, M. and Smale, S., ‘Complexity of Bézout’s Theorem III: condition number and packing’, J. Complexity 9 (1993), 414.
[53]Shub, M. and Smale, S., ‘Complexity of Bézout’s Theorem IV: probability of success; extensions’, SIAM J. Numer. Anal. 33 (1996), 128148.
[54]Shub, M. and Smale, S., ‘Complexity of Bézout’s Theorem V: polynomial time’, Theor. Comput. Sci. 133 (1994), 141164.
[55]Smale, S., ‘Some remarks on the foundations of numerical analysis’, SIAM Rev. 32 (1990), 211220.
[56]Smale, S., ‘Complexity theory and numerical analysis’, in: Acta Numerica (ed. Iserles, A.) (Cambridge University Press, Cambridge, UK, 1997), 523551.
[57]Smale, S., ‘Mathematical problems for the next century’, in: Mathematics: Frontiers and Perspectives (eds. Arnold, V., Atiyah, M., Lax, P. and Mazur, B.) (American Mathematical Society, Providence, RI, 2000), 271294.
[58]Trakhtenbrot, B. A., ‘A survey of russian approaches to perebor (brute-force search) algorithms’, Ann. Hist. Comput. 6 (1984), 384400.
[59]Turing, A. M., ‘Rounding-off errors in matrix processes’, Quart. J. Mech. Appl. Math. 1 (1948), 287308.
[60]von Neumann, J. and Goldstine, H. H., ‘Numerical inverting matrices of high order’, Bull. Amer. Math. Soc. 53 (1947), 10211099.
[61]von Neumann, J. and Goldstine, H. H., ‘Numerical inverting matrices of high order, II’, Proc. Amer. Math. Soc. 2 (1951), 188202.
[62]Weihrauch, K., Computable Analysis, Texts in Theoretical Computer Science. An EATCS Series, (Springer, Berlin, 2000).
[63]Wilkinson, J., ‘Some comments from a numerical analyst’, J. ACM 18 (1971), 137147.
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