## Linear programming in linear time when the dimension is fixed (1984)

Venue: | J. ACM |

Citations: | 198 - 12 self |

### BibTeX

@ARTICLE{Megiddo84linearprogramming,

author = {Nimrod Megiddo},

title = {Linear programming in linear time when the dimension is fixed},

journal = {J. ACM},

year = {1984},

volume = {31},

pages = {114--127}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. It is demonstrated that the linear programming problem in d variables and n constraints can be solved in O(n) time when d is fixed. This bound follows from a multidimensional search technique which is applicable for quadratic programming as well. There is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n. Categories and Subject Descriptors: F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problems-computations on matrices; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems-geometrical problems and computations; sort-ing and searching; G. 1.6 [Mathematics of Computing]: Optimization-linear programming

### Citations

470 | Computational Geometry
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...Given n points a, = (a,l, . . . , u,d) E Rd (i = 1, . . . , n), we wish to find a linear function so as to minimize max( 1 c$,' a,a,, + a d - u,d I : i = 1, . . . , n). For other related problems see =-=[18]-=-. We also note that the results can be extended to solve minimization convex quadratic programming problems. For the details of the extension to quadratic programming, as well as other related problem... |

391 | Time bounds for selection
- Blum, Floyd, et al.
- 1973
(Show Context)
Citation Context ...ce (where the expectation is relative to our random choices). The analysis is close to that of algorithm FIND (see [9]). Another approach is to employ a probabilistic selection algorithm like that in =-=[5]-=-, but, again, it is not required that the exact median be found. Finally, we remark that a hybrid multidimensional search (i.e., picking, recursively, the better between the two approaches whenever a ... |

269 | Convex polytopes
- GruÌˆnbaum
- 1967
(Show Context)
Citation Context ...f variables and n is the number of constraints). We call such an algorithm genuinely polynomial. This question is closely related to other interesting open questions in the theory of convex polytopes =-=[6]-=- concerning the diameter, height, etc., of polytopes. Obviously, Khachian's results has not advanced our knowledge about these problems. With the central question still open, it is interesting to inve... |

256 |
Mathematics for the analysis of algorithms
- Greene, Knuth
- 1990
(Show Context)
Citation Context ...is is repeated independently many times, we do achieve expected linear-time performance (where the expectation is relative to our random choices). The analysis is close to that of algorithm FIND (see =-=[9]-=-). Another approach is to employ a probabilistic selection algorithm like that in [5], but, again, it is not required that the exact median be found. Finally, we remark that a hybrid multidimensional ... |

202 | Linear-time algorithms for linear programming in R3 and related problems - Megiddo - 1983 |

178 |
How good is the Simplex algorithm
- Klee, Minty
- 1972
(Show Context)
Citation Context ...lgorithm is not). However, the crude bound of O(nd) (assuming d < n) means that no matter how slowly the dimension grows, we have a superpolynomial bound. Even the more refined bound of O(nC''') (see =-=[8]-=-) does not help in this respect. Here we develop an algorithm that is genuinely polynomial even if d grows slowly with n. In this paper we study the complexity of linear programming in a fixed space. ... |

169 | Pattern Classification and Scene Analysis. WileyInterscience - Duda, Hart - 1973 |

126 |
Multidimensional divide-and-conquer
- Bentley
- 1980
(Show Context)
Citation Context ..., two problems each with n/2 objects in (d - 1) space and then our problem is reduced to one with n/2 objects in d space. This should not be confused with Bentley's multidimensional divideand-conquer =-=[2]-=-, where an n x d problem is reduced to solving two (1112) x d-problems and then one n x (d - 1) problem. We note that even though the results of this paper are interesting from the theoretical point o... |

117 |
A polynomial algorithm in linear programming (in Russian). Doklady Adademiia Nauk SSSR
- Khachian
- 1979
(Show Context)
Citation Context ...f linear programminghs one of the questions that has attracted many researchers since the invention of the simplex algorithm [3]. A major theoretical development in the field was Khachian's algorithm =-=[7]-=-, which proved that the problem could be solved in time that is polynomial in the sum of logarithms of the (integral) coefficients. This notion of polynomiality is not altogether satisfactory (see [lo... |

65 | Finding the median - Schonhage, Paterson, et al. - 1976 |

29 |
Towards a genuinely polynomial algorithm for linear programming
- Megiddo
- 1983
(Show Context)
Citation Context ...till open, it is interesting to investigate special classes of linear programming problems. Systems of linear inequalities with no more than two variables per inequality have been shown by the author =-=[13]-=- to have a genuinely polynomial algorithm. When either n or d is fixed, then the simplex algorithm is This work was supported in part by the National Science Foundation under Grants ECS-8 12 174 1 and... |

7 |
On the average speed of the simplex method in linear programming
- Smale
- 1983
(Show Context)
Citation Context ...complexity of the simplex algorithm as the number of variables tends to infinity while the number of constraints is fixed. The answer is highly sensitive to a probabilistic model to be adopted. Smale =-=[19]-=- has shown that, under a certain model, the average number of pivot steps is o(nf) for every t > 0 whenever the number of constraints is fixed. Using Approach 11, we reduce a problem of order n x d to... |

6 |
The Design andAnalysis of Computer Algorithms.Addision
- Aho, croft, et al.
- 1974
(Show Context)
Citation Context ...re solvable by this technique. Current computational experience with d = 3 looks very successful. Direct applications of linear programming in which the dimension is normally small are the following: =-=(1)-=- Linear separability. Given n points in Rd, organized in two disjoint sets, find a hyperplane (if there is one) that separates the two sets. This problem is useful in statistics and in pattern recogni... |

4 |
Combinatorial Solutions of Multidimensional Divide-and-Conquer Recurrences
- Monier
- 1980
(Show Context)
Citation Context ..., 2) = 26, Ql(32, 2) = 3 1. If n = 2L where L is an integer, then instead of Q, we may consider a recurrence of the form with boundary conditions F(L, 1) = L + 1 and F(0, d) = 1. The solution is (see =-=[15]-=- for an interesting solution method): It follows that F(L, d) < (2~)~/(d - 2)! so that for fixed d we have Q(n, d) = O(logdn) with a surprisingly favorable constant C = C(d) = 2d/(d - 2)!.sLinmr Progr... |

3 |
The approximation of functions, vol. I, linear theory. Reading Mass
- Rice
- 1964
(Show Context)
Citation Context ..., organized in two disjoint sets, find a hyperplane (if there is one) that separates the two sets. This problem is useful in statistics and in pattern recognition [4, 141. (2) Chebyshev approximution =-=[16]-=-. Given n points a, = (a,l, . . . , u,d) E Rd (i = 1, . . . , n), we wish to find a linear function so as to minimize max( 1 c$,' a,a,, + a d - u,d I : i = 1, . . . , n). For other related problems se... |

1 |
Linear Programming and Extensions
- DANTLIG
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Citation Context ...blem, linear time algorithms 1. Introduction The computational complexity of linear programminghs one of the questions that has attracted many researchers since the invention of the simplex algorithm =-=[3]-=-. A major theoretical development in the field was Khachian's algorithm [7], which proved that the problem could be solved in time that is polynomial in the sum of logarithms of the (integral) coeffic... |

1 | binary encoding appropriate for the problem-language relationship? Theor - MEGIDW - 1982 |

1 | Computer-OrientedApproaches to Pattern Recognition - MEISEL |