# Two New Bounds for the Random‐Edge Simplex‐Algorithm

## Abstract

*d*‐polytope with

*n*vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial

*d*‐cubes the trivial upper bound of $2^d$ on the performance of RANDOM‐EDGE can asymptotically be improved by the factor $1/d^{(1-\varepsilon)\log d}$ for every $\varepsilon>0$.

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## References

*Math. Oper. Res.*, 1 (1976), pp. 89–95.

*Deformed products and maximal shadows of polytopes*, in Advances in Discrete and Computational Geometry, Contemp. Math. 223, J. Chazelle, J. E. Goodman, and R. Pollack, eds., Amer. Math. Soc., Providence, RI, 1999, pp. 57–90.

*The Klee‐Minty Random Edge Chain Moves With linear Speed*, manuscript, 2004.

*Combinatorial Aspects of Convex Polytopes*, in Handbook of Convex Geom. A, B, North–Holland, Amsterdam, 1993, pp. 485–534.

*The Simplex Method: A Probabilistic Analysis*, in Algorithms Combin., Springer‐Verlag, New York, 1987.

*Inform. Process. Lett.*, 56 (1995), pp. 79–81.

*Linear Programming*, W. H. Freeman, New York, 1983.

*Linear Programming and Extensions*, Princeton University Press, Princeton, New Jersey, 1963.

*n*points,

*Random Structures Algorithms*, 23 (2003), pp. 453–471.

*Concrete Mathematics*, 2nd ed., Addison‐Wesley, Reading, MA, 1994.

*Probability and Random Processes*, Clarendon Press, London, 1982.

*A proof of the strict monotone 4‐step conjecture*, in Advances in Discrete and Computational Geometry, Contemp. Math. 233, J. Chazelle, J. E. Goodman, and R. Pollack, eds., Amer. Math. Soc., Providence, RI, 1998, pp. 201–216.

*SIAM J. Comput.*, 34 (2005), pp. 475–497.

*A subexponential randomized simplex algorithm*, Random Structures Algorithms, to appear.

*Math. Program.*, 79 (1997), pp. 217–233.

*Combinatorica*, 4 (1984), pp. 373–395.

*Sov. Math. Doklady*, 20 (1979), pp. 1093–1096.

*A randomized polynomial‐time simplex algorithm for linear programming*, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 51–60.

*How Good is the Simplex Algorithm?*, in Inequalities III, O. Shisha, ed., Academic Press, New York, 1972, pp. 159–175.

*Random Structures Algorithms*, 5 (1994), pp. 591–607.

*Algorithmica*, 16 (1996) pp. 498–516.

*Adv. Math.*, 204 (2006), pp. 262–277.

*Theory of Linear and Integer Programming*, Wiley‐Interscience, New York, 1986.

*J. ACM*, 51 (2004), pp. 385–463.

*Math. Program.*, 35 (1986), pp. 193–224.

*Congr. Numer.*, 50 (1985), pp. 165–169.

*Discrete Appl. Math.*, 20 (1988), pp. 69–81.

*Lectures on Polytopes*, Grad. Texts in Math. 152, Springer‐Verlag, Heidelberg, 1994.

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#### History

**Submitted**: 1 February 2005

**Accepted**: 27 September 2006

**Published online**: 15 March 2007

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