Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time

Journal of the ACM

Volumn 51, Issue 3, 2004, Pages 385-463

  Spielman, Daniel A ^a   Teng, Shang Hua ^b,c  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b BOSTON UNIVERSITY   (United States)

^c AKAMAI TECHNOLOGIES   (United States)

Author keywords

Complexity; Perturbation; Simplex method; Smoothed analysis

Indexed keywords

AVERAGE-CASE ANALYSES; SIMPLEX METHOD; SMOOTHED ANALYSIS;

ALGORITHMS; COMPUTATIONAL METHODS; HEURISTIC METHODS; INTERPOLATION; PERTURBATION TECHNIQUES; STATISTICAL METHODS;

POLYNOMIALS;

EID: 4243066295     PISSN: 00045411     EISSN: None     Source Type: Journal    
DOI: 10.1145/990308.990310     Document Type: Article

References (45)

1. ABRAMOWITZ, M., AND STEGUN, I. A., Eds. 1970. Handbook of Mathematical Functions, 9 ed. Applied Mathematics Series, vol. 55. National Bureau of Standards.
2. ADLER, I. 1983. The expected number of pivots needed to solve parametric linear programs and the efficiency of the self-dual simplex method. Tech. Rep., University of California at Berkeley. May.
3. 2)) expected number of pivot steps. J. Complex. 3, 372-387.
4. ADLER, I., AND MEGIDDO, N. 1985. A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension. J. ACM 32, 4 (Oct.), 871-895.
5. AMENTA, N., AND ZIEGLER, G. 1999. Deformed products and maximal shadows of polytopes. In Advances in Discrete and Computational Geometry, B. Chazelle, J. Goodman, and R. Pollack, Eds. Number 223 in Contemporary Mathematics. American Mathematics Society, 57-90.
6. AVIS, D., AND CHVÁTAL, V. 1978. Notes on Bland's pivoting rule. In Polyhedral Combinatorics, Math. Programming Study, vol. 8. 24-34.
7. BLASCHKE, W. 1935, Integralgeometrie 2: Zu ergebnissen von M. W. Crofton. Bull. Math. Soc. Roumaine Sci. 37, 3-11.
8. BLUM, A., AND SPENCER, J. 1995. Coloring random and semi-random k-colorable graphs. J. Algorithms 19, 2, 204-234.
9. BORGWARDT, K. H. 1977. Untersuchungen zur asymptotik der mittleren schrittzahl von simplexverfahren in der linearen optimierung. Ph.D. dissertation, Universitat Kaiserslautern.
10. BORGWARDT, K. H. 1980. The Simplex Method: a probabilistic analysis. Number 1 in Algorithms and Combinatorics. Springer-Verlag, New York.
11. CHAN, T. F., AND HANSEN, P. C. 1992. Some applications of the rank revealing QR factorization. SIAM J. Sci. Stat. Comput. 13, 3 (May), 727-741.
12. DANTZIG, G. B. 1951. Maximization of linear function of variables subject to linear inequalities. In Activity Analysis of Production and Allocation, T. C. Koopmans, Ed. 339-347.
13. EDELMAN, A. 1992. Eigenvalue roulette and random test matrices. In Linear Algebra for Large Scale and Real-Time Applications, M. S. Moonen, G. H. Golub, and B. L. R. D. Moor, Eds. NATO ASI Series. 365-368.
14. EFRON, B. 1965. The convex hull of a random set of points. Biometrika 52, 3/4, 331-343.
15. FEIGE, U., AND KILIAN, J. 1998. Heuristics for finding large independent sets, with applications to coloring semi-random graphs. In 39th Annual Symposium on Foundations of Computer Science: Proceedings (Palo Alto, Calif. Nov. 8-11). IEEE Computer Society Press, Los Alamitos, Calif.
16. FEIGE, U., AND KRAUTHGAMER, R. 1998. Improved performance guarantees for bandwidth minimization heuristics. http://www.cs.berkeley.edu/~robi/papers/FK- BandwidthHenristics-manuscript98.ps.gz.
17. FELLER, W. 1968. An Introduction to Probability Theory and Its Applications. Vol. 1. Wiley, New York.
18. FELLER, W. 1971. An Introduction to Probability Theory and Its Applications. Vol. 2. Wiley, New York.
19. GOLDFARB, D. 1983. Worst case complexity of the shadow vertex simplex algorithm. Tech. Rep. Columbia Univ., New York.
20. GOLDFARB, D., AND SIT, W. T. 1979. Worst case behaviour of the steepest edge simplex method. Disc. Appl. Math 1, 277-285.
21. HAIMOVICH, M. 1983. The simplex algorithm is very good!: On the expected number of pivot steps and related properties of random linear programs. Tech. Rep. Columbia Univ., New York.
22. JEROSLOW, R. G. 1973. The simplex algorithm with the pivot rule of maximizing improvement criterion. Disc. Math. 4, 367-377.
23. KALAI, G. 1992. A subexponential randomized simplex algorithm. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing (Victoria, B.C., Canada). ACM, New York, 475-482.
24. KALAI, G., AND KLEITMAN, D. J. 1992. A quasi-polynomial bound for the diameter of graphs of polyhedra. Bull. Amer. Math. Soc. 26, 315-316.
25. KARMARKAR, N. 1984. A new polynomial time algorithm for linear programming. Combinatorica 4, 373-395.
26. KHACHIYAN, L. G. 1979. A polynomial algorithm in linear programming. Doklady Akademia Nauk SSSR, 1093-1096.
27. KLEE, V., AND MINTY, G. J. 1972. How good is the simplex algorithm? In Inequalities - III, Shisha, O., Ed. Academic Press, Orlando, Fla., 159-175.
28. MATOUŠEK, J., SHARIR, M., AND WELZL, E. 1996. A subexponential bound for linear programming. Algorithmica 16, 4/5 (Oct./Nov.), 498-516.
29. MEGIDDO, N. 1986. Improved asymptotic analysis of the average number of steps performed by the self-dual simplex algorithm. Math. Prog. 35, 2, 140-172.
30. MILES, R. E. 1971. Isotropic random simplices. Adv. Appl. Prob. 3, 353-382.
31. MURTY, K. G. 1980. Computational complexity of parametric linear programming. Math. Prog. 19, 213-219.
32. RENEGAR, J. 1994. Some perturbation theory for linear programming. Math. Prog. Ser. A. 65, 1, 73-91.
33. RENEGAR, J. 1995a. Incorporating condition measures into the complexity theory of linear programming. SIAM J. Optim. 5, 3, 506-524.
34. RENEGAR, J. 1995b. Linear programming, complexity theory and elementary functional analysis. Math. Prog. Ser. A. 70, 3, 279-351.
35. RENYI, A., AND SULANKE, R. 1963. Uber die convexe hulle von is zufallig gewahlten punkten, I. Z. Whar. 2, 75-84.
36. RENYI, A., AND SULANKE, R. 1964. Uber die convexe hulle von is zufallig gewahlten punkten, II. Z. Whar. 3, 138-148.
37. SANTALO, L. A. 1976. Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and its Applications. Addison-Wesley, Reading, Mass.
38. SANTHA, M., AND VAZIRANI, U. 1986. Generating quasi-random sequences from semi-random sources. J. Comput. System Sciences 33, 75-87.
39. SMALE, S. 1982. The problem of the average speed of the simplex method. In Proceedings of the 11th International Symposium on Mathematical Programming. 530-539.
40. SMALE, S. 1983. On the average number of steps in the simplex method of linear programming. Math. Prog. 27, 241-262.
41. TODD, M. 1986. Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems. Math. Prog. 35, 173-192.
42. TODD, M. J. 1991. Probabilistic models for linear programming. Math. Oper. Res. 16, 4, 671-693.
43. TODD, M. J., TUNÇEL, L., AND YE, Y. 2001. Characterizations, bounds, and probabilistic analysis of two complexity measures for linear programming problems. Math. Program., Ser. A 90, 59-69.
44. TURNER, J. S. 1986. On the probable performance of a heuristic for bandwidth minimization. SIAM J. Comput. 15, 2, 561-580.
45. VAVASIS, S. A., AND YE, Y. 1996. A primal-dual interior-point method whose running time depends only on the constraint matrix. Math. Program. 74, 79-120.