Jonathan A. Kelner
ABSTRACT
We present the first randomized polynomial-time simplex algorithm for linear programming. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input.We begin by reducing the input linear program to a special form in which we merely need to certify boundedness. As boundedness does not depend upon the right-hand-side vector, we run the shadow-vertex simplex method with a random right-hand-side vector. Thus, we do not need to bound the diameter of the original polytope.Our analysis rests on a geometric statement of independent interest: given a polytope A x ≤ b in isotropic position, if one makes a polynomially small perturbation to b then the number of edges of the projection of the perturbed polytope onto a random 2-dimensional subspace is expected to be polynomial.
- D. Bertsimas and S. Vempala. Solving convex programs by random walks. J. ACM, 51(4):540--556, 2004. Google Scholar
Digital Library
- K. H. Borgwardt. The Simplex Method: a probabilistic analysis. Number 1 in Algorithms and Combinatorics. Springer-Verlag, 1980.Google Scholar
- G. B. Dantzig. Maximization of linear function of variables subject to linear inequalities. In T. C. Koopmans, editor, Activity Analysis of Production and Allocation, pages 339--347. 1951.Google Scholar
- G. B. Dantzig. Linear Programming and Extensions. Princeton University Press, 1963.Google ScholarDigital Library
- A. Deshpande and D. A. Spielman. Improved smoothed analysis of the shadow vertex simplex method. preliminary version appeared in FOCS '05, 2005. Google ScholarDigital Library
- J. Dunagan and S. Vempala. A simple polynomial-time rescaling algorithm for solving linear programs. In Proceedings of the thirty-sixth annual ACM Symposium on Theory of Computing (STOC-04), pages 315--320, New York, June 13--15 2004. ACM Press. Google ScholarDigital Library
- K. Fukuda and T. Terlaky. Criss-cross methods: A fresh view on pivot algorithms. Mathematical Programming, 79:369--395, 1997. Google ScholarDigital Library
- S. Gass and T. Saaty. The computational algorithm for the parametric objective function. Naval Research Logistics Quarterly, 2:39--45, 1955.Google ScholarCross Ref
- M. Grotschel, L. Lovasz, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, 1991.Google Scholar
- G. Kalai and D. J. Kleitman. A quasi-polynomial bound for the diameter of graphs of polyhedra. Bulletin Amer. Math. Soc., 26:315--316, 1992.Google ScholarCross Ref
- N. Karmarkar. A new polynomial time algorithm for linear programming. Combinatorica, 4:373--395, 1984. Google ScholarDigital Library
- L. G. Khachiyan. A polynomial algorithm in linear programming. Doklady Akademia Nauk SSSR, pages 1093--1096, 1979.Google Scholar
- N. Megiddo and R. Chandrasekaran. On the epsilon-perturbation method for avoiding degeneracy. Operations Research Letters, 8:305--308, 1989.Google ScholarDigital Library
- A. Schrijver. Theory of Linear and Integer Programming. John Wiley & Sons, 1986. Google ScholarDigital Library
- D. A. Spielman and S.-H. Teng. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. JACM: Journal of the ACM, 51:385 -- 463, 2004. Google ScholarDigital Library
Index Terms
- A randomized polynomial-time simplex algorithm for linear programming
Recommendations
A simplex algorithm for piecewise-linear programming I: Derivation and proof
The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. This three-part paper develops and analyzes a general, computationally practical ...
On linear programming relaxations for solving polynomial programming problems
Highlights- Theoretically compared the tightness of the J-set relaxation with the recursive McCormick relaxations.
AbstractThis paper studies linear programming (LP) relaxations for solving polynomial programming problems. A polynomial programming problem can be equivalently formulated as a quadratically constrained quadratic program (QCQP) by introducing ...
A simplex algorithm for piecewise-linear programming II: Finiteness, feasibility and degeneracy
The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. Part I of this paper has developed a general and direct simplex algorithm for ...
Comments