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Deterministic and randomized polynomial-time approximation of radii

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  26 February 2010

Andreas Brieden ,
Peter Gritzmann ,
Ravindran Kannan ,
Victor Klee ,
László Lovász  and
Miklós Simonovits
Andreas Brieden
Affiliation:
Zentrum Mathematik, Technische Universitôt München, D-80290 Munich, Gernmany E-mail: brieden@mathematik.tu-muenchen.de
Peter Gritzmann
Affiliation:
Zentrum Mathematik, Technische Universitôt München, D-80290 Munich, Gernmany E-mail: brieden@mathematik.tu-muenchen.de
Ravindran Kannan
Affiliation:
Department of Computer Science, Yale University, New Haven. CT 06520, U.S.A. E-mail: kannan@cs.yale.edu
Victor Klee
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, U.S.A. E-mail: klee@math.washington.edu
László Lovász
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98053, U.S.A. E-mail: lovasz@microsoft.com
Miklós Simonovits
Affiliation:
Alfréd Renyi Institute of Mathematics, Reáltanoda u. 13-15, H-1053, Budapest, Hungary E-mail: miki@math-inst.hu

Abstract

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This paper is concerned with convex bodies in n-dimensional lp, spaces, where each body is accessible only by a weak separation or optimization oracle. It studies the asymptotic relative accuracy, as n→∞, of polynomial-time approximation algorithms for the diameter, width, circumradius, and inradius of a body K, and also for the maximum of the norm over K.

MSC classification

Secondary: 52B55: Computational aspects related to convexity
Type
Research Article
Information
Mathematika , Volume 48 , Issue 1-2 , December 2001 , pp. 63 - 105
Copyright
Copyright © University College London 2001

References

[BF87]Bárány, I. and Füredi, Z.. Computing the volume is difficult. Discrete Comput. Geom.,2 (1987), 319–326.CrossRefGoogle Scholar
[BF88]Bárány, I. and Füredi, Z.. Approximation of the sphere by polytopes having few vertices. Proc. Amer. Math. Soc, 102 (1988), 651–659.Google Scholar
[BG97]Brieden, A. and Gritzmann, P.. On Helly's theorem: algorithms and extensions. Discrete Comput. Geom., 17 (1997), 393–410.CrossRefGoogle Scholar
[BGK+98]Brieden, A., Gritzmann, P., Kannan, R., Klee, V., Lovász, L. and Simonovits, M..Approximation of radii and norm-maxima: Randomization doesn't help. Proc. 39th Sympos. FOCS, IEEE (1998), 244–251.Google Scholar
[BGK99]Brieden, A., Gritzmann, P. and Klee, V.. Inapproximability of some geometric and quadratic optimization problems. Approximation and Complexity in Numerical Optimization (ed. Pardalos, P. M.). Kluwer (Boston, 2000), 96–115.Google Scholar
[BGKL90]Bodlaender, H. L., Gritzmann, P., Klee, V. and van Leeuwen, J.. Computational complexity of norm-maximization. Combinatorica, 10 (1990), 203–225.CrossRefGoogle Scholar
[BP90]Ball, K. and Pajor, A.. Convex bodies with few faces. Proc. Amer. Math. Soc, 110 (1990), 225–231.Google Scholar
[Bri98]Brieden, A.. Deterministic Approximation Algorithms in Computational Convexity, PhD thesis (Tech. Univ. Munich, 1998).Google Scholar
[Car85]Carl, B.. Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces. Ann. I'Inst. Fourier, 35 (1985), 79–118.CrossRefGoogle Scholar
[CP88]Carl, B. and Pajor, A.. Gelfand numbers of operators with values in a Hilbert space.Invent. Math., 94 (1988), 479–504.Google Scholar
[DF88]Dyer, M. and Frieze, A.. On the complexity of computing the volume of a polytope.SIAM J. Comput., 17 (1988), 967–974.CrossRefGoogle Scholar
[DFK89]Dyer, M., Frieze, A. and Kannan, R.. A random polynomial time algorithm for approximating the volume of convex bodies. Proc. 21st ACM Sympos. Theory Comput. (1989), 375–381.CrossRefGoogle Scholar
[DGK63]Danzer, L., Grünbaum, B. and Klee, V.. Helly's theorem and its relatives. Proc. Sympos.Pure Math., 1 (1963), 101–180.Google Scholar
[Dud94]Dudov, S. I.. An inner bound of a convex set by a norm body. Comput. Math. Math. Physics, 36 (1994), 683–688.Google Scholar
[Ele86]Elekes, G.. A geometric inequality and the complexity of computing volume. Discrete Comput. Geom., 1 (1986), 289–292.Google Scholar
[FO85]Freund, R. M. and Orlin, J. B.. On the complexity of four polyhedral set containment problems. Math. Programming, 33 (1985), 139–145.CrossRefGoogle Scholar
[Gil77]Gill, J.. Computational complexity of probabilistic Turing machines. SIAM J. Corn-put., 6 (1977), 675–695.Google Scholar
[GJ79]Garey, M. R. and Johnson, D. S.. Computers and Intractability. A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979).Google Scholar
[GK92]Gritzmann, P. and Klee, V.. Inner and outer j-radii of convex bodies in finite-dimensional normed spaces. Discrete Comput. Geom., 7 (1992), 255–280.Google Scholar
[GK93]Gritzmann, P. and Klee, V.. Computational complexity of inner and outer j-radii of polytopes in finite-dimensional normed spaces. Math. Programming, 59 (1993), 163–213.Google Scholar
[GK94]Gritzmann, P. and Klee, V.. On the complexity of some basic problems in computational convexity: I. Containment problems. Discrete Math., 136 (1994), 129–174.Google Scholar
[GLS93]Grötschel, M., Lovász, L. and Schrijver, A.. Geometric Algorithms and CombinatorialOptimization (Springer, Berlin, 1988, 1993).CrossRefGoogle Scholar
[GMR94]Gordon, Y., Meyer, M. and Reisner, S.. Volume approximation of convex bodies—a constructive method. Studia Math., 111 (1994) 81–95.Google Scholar
[GMR95]Gordon, Y., Meyer, M. and Reisner, S.. Constructing a polytope to approximate a convex body. Geom. Dedicata, 57 (1995), 217–222.Google Scholar
[Gru93]Gruber, P. M.. Aspects of approximation of convex bodies. Handbook of Convex Geometry. Gruber, P. M. and Wills, J. M., eds. (Elsevier, Amsterdam, 1993), 319–345.Google Scholar
[HJ74]Hoffmann-Jørgensen, J.. Sums of independent Banach space valued random variables. Studia Math., 52 (1974), 159–186.CrossRefGoogle Scholar
[HJP76]Hoffman-Jørgensen, J. and Pisier, G.. The law of large numbers and the central limit theorem in Banach spaces. Ann. Probab., 4 (1976), 587–599.Google Scholar
[Jan98]Jansen, T.. Introduction to the theory of complexity and approximation algorithms. In Mayr et al. [MPS98], 5–28.Google Scholar
[JunOl]Jung, H. W. E.. Über den kleinsten Kreis, der eine ebene Figur einschlieBt. J. Reine Angew. Math., 130 (1901), 310–313.Google Scholar
[JVV86]Jerrum, M. R.,Valiant, L. G. and Vazirani, V. V.. Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci., 43 (1986), 169–188.Google Scholar
[Kah68]Kahane, J.-P.. Some Random Series of Functions (Heath, Cambridge, 1968; 2nd edition, Cambridge University Press, 1985).Google Scholar
[Kha79]Khachiyan, L. G.. Polynomial algorithms in linear programming. Dokl. Akad. Nauk SSSR, 244 (1979), 1093–1096. (Russian); English translation: Doklady Mathematics, 20 (1979), 191–194.Google Scholar
[Kha88]Khachiyan, L. G.. On the complexity of computing the volume of a polytope. Izv. Akad. Nauk SSSR, Engineering Cybernetics, 3 (1988), 216–217.Google Scholar
[Khi23]Khintchine, A.. Über dyadische Brüche. Math. Z., 18 (1923), 109–116.Google Scholar
[KLS98]Kannan, R., Lovász, L. and Simonovits, M.. Random walks and an O *(n 5) volume algorithm for convex bodies. Random Structures Algorithms, 11 (1998), 1–50.Google Scholar
[Koc94]Kochol, M.. Constructive approximation of a ball by polytopes. Math. Slovaca, 44 (1994), 99–105.Google Scholar
[LS92]Lovász, L. and Simonovits, M.. On the randomized complexity of volume and diameter. Proc. 33rd Sympos. FOCS, IEEE (1992), 482–491.CrossRefGoogle Scholar
[LT79]Lindenstrauss, J. and Tzafriri, L.. Classical Banach Spaces II(Springer, Berlin, 1979).Google Scholar
[MPS98]Mayr, E. W., Prómel, H. J. and Steger, A. (eds.). Lectures on Proof Verification and Approximation Algorithms. Lecture Notes in Computer Science, no. 1367 (Springer, 1998).CrossRefGoogle Scholar
[MS86]Milman, V. D. and Schechtman, G.. Asymptotic Theory of Finite Dimensional Normed Spaces. Lecture Notes in Mathematics, vol. 1200 (Springer, 1986).Google Scholar
[Pie80]Pietsch, A.. Operator Ideals (North-Holland, Amsterdam, 1980).Google Scholar
[Pis88]Pisier, G.. The Volume of Convex Bodies and Banach Space Geometry (Cambridge University Press, 1988).Google Scholar
[PY91]Papadimitriou, C. H. and Yannakakis, M.. Optimization, approximation, and complexity classes. J. Comput. System Sci., 43 (1991), 425–440.Google Scholar
[Sho77]Shor, N. Z.. Cut-off method with space extension in convex programming problems. Kibemetika, 1977 (1977), no. 1, 94–95; English translation: Cybernetics, 13 (1977), 94–96.Google Scholar
[Ste22]Steinhagen, P.. Über die gröBte Kugel in einer konvexen Punktmenge. Abh. Math. Sem. Univ. Hamburg (1922), 15–22.Google Scholar
[SY92]Seberry, J. R. and Yamada, M.. Hadamard matrices, sequences, and block designs. Contemporary Design Theory-A Collection of Surveys (North Holland, New York, 1992), 431–560.Google Scholar
[Syl67]Sylvester, J. J.. Thoughts on inverse orthogonal matrices, simultaneous sign-successions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile work, and the theory of numbers. Philos. Magazine, 34 (1867), 461–475.Google Scholar
[TJ88]Tomczak-Jaegermann, N.. Banach-Mazur Distances and Finite-Dimensional Operator Ideals (Longman Scientific & Technical, 1988).Google Scholar
[Vaa79]Vaaler, J. D.. A geometric inequality with applications to linear forms. Pacific J. Math., 83 (1979), 543–553.Google Scholar
[YN76]Yudin, B. D. and Nemirovskii, A. S.. Informational complexity and efficient methods for the solution of convex extremal problems. Ekonomika i Mat. Metody, 12 (1976), 357–369; English translation: Matekon, 13, 25–45.Google Scholar