## Maximizing Concave Functions in Fixed Dimension (1993)

Venue: | in: Complexity in Numeric Computation |

Citations: | 13 - 0 self |

### BibTeX

@INPROCEEDINGS{Cohen93maximizingconcave,

author = {Edith Cohen and Nimrod Megiddo},

title = {Maximizing Concave Functions in Fixed Dimension},

booktitle = {in: Complexity in Numeric Computation},

year = {1993},

pages = {74--87},

publisher = {World Scientific Press}

}

### OpenURL

### Abstract

In [3, 5, 2] the authors introduced a technique which enabled them to solve the parametric minimum cycle problem with a fixed number of parameters in strongly polynomial time. In the current paper 1 we present this technique as a general tool. In order to allow for an independent reading of this paper, we repeat some of the definitions and propositions given in [3, 5, 2]. Some proofs are not repeated, however, and instead we supply the interested reader with appropriate pointers. Suppose Q ae R d is a convex set given as an intersection of k halfspaces, and let g : Q ! R be a concave function that is computable by a piecewise affine algorithm (i.e., roughly, an algorithm that performs only multiplications by scalars, additions, and comparisons of intermediate values which depend on the input). Assume that such an algorithm A is given and the maximal number of operations required by A on any input (i.e., point in Q) is T . We show that under these assumptions, for any fixed d, the ...

### Citations

193 | Linear programming in linear time when the dimension is fixed
- Megiddo
- 1984
(Show Context)
Citation Context ..., the decision can be made by considering a neighborhood of the maximum of the function relative to H 0 , searching for a direction of ascent from that point. This principle is explained in detail in =-=[11]. For-=- a hyperplane H 0 ae R d , we wish to decide on which side of H 0 the set rel intslies. By solving a linear program with d variables and k +1 constraints, we determine whether or not H 0 " Q = ;,... |

60 |
On a multidimensional search technique and its application to the Euclidean one-center problem
- Dyer
- 1986
(Show Context)
Citation Context ...her (i) O(fl(d \Gamma 1) log 2 s) parallel time on O(s) processors, or (ii) O(fl(d \Gamma 1)s log s) sequential time. The function fl(d) arises from the multi-dimensional search [11]. It follows from =-=[1, 8]-=- that fl(d) = 3 O(d 2 ) . 5. The algorithm The algorithm described below solves Problem 2.8. It finds a vectors2 rel int , unless g is unbounded. It also returns a collection C of pieces of g whose mi... |

29 |
Towards a genuinely polynomial algorithm for linear programming
- Megiddo
- 1983
(Show Context)
Citation Context ...wo variables per inequality. Linear programming problems with at most two variables in each constraint and in the objective function were shown to have a strongly polynomial time algorithm by Megiddo =-=[10]-=-. Lueker, Megiddo and Ramachandran [9] gave a polylogarithmic time parallel algorithm for the problem which uses a quasipolynomial number of processors. The best known time bounds for the problem were... |

27 |
Improved algorithms for linear inequalities with two variables per inequality
- Cohen, Megiddo
- 1994
(Show Context)
Citation Context ...Megiddo and Ramachandran [9] gave a polylogarithmic time parallel algorithm for the problem which uses a quasipolynomial number of processors. The best known time bounds for the problem were given in =-=[7, 2]-=-. Cosares, using nested parametrization, extended Megiddo's strong polynomiality result to allow objective functions which have a fixed number of nonzero coefficients. This result can be further exten... |

18 | Using separation algorithms in fixed dimension
- Norton, Plotkin, et al.
- 1992
(Show Context)
Citation Context ... defined with respect to the lexicographic order as discussed in the introduction. Below we present some applications of Theorem 8.2. Additional applications were found by Norton, Plotkin, and Tardos =-=[12]-=-. Adding variables to LP's with two variables per inequality. Linear programming problems with at most two variables in each constraint and in the objective function were shown to have a strongly poly... |

17 |
Linear programming
- Clarkson
- 1986
(Show Context)
Citation Context ...rformed in either (i) O( (d ; 1) log 2 s) parallel time on O(s) processors, or (ii) O( (d ; 1)s log s) sequential time. The function (d) arises from the multi-dimensional search [11]. It follows from =-=[1, 8]-=- that (d) =3 O(d2 ) . 5. The algorithm The algorithm described below solves Problem 2.8. It nds a vector 2 rel int , unless g is unbounded. It also returns a collection C of pieces of g whose minimum ... |

5 |
Combinatorial Algorithms for Optimization Problems
- Cohen
- 1991
(Show Context)
Citation Context ..., CA 94305 and IBM Almaden Research Center. Nimrod Megiddo IBM Almaden Research Center, San Jose, CA 95120-6099 and School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel. Abstract In =-=[3, 5, 2]-=- the authors introduced a technique which enabled them to solve the parametric minimum cycle problem with a fixed number of parameters in strongly polynomial time. In the current paper 1 we present th... |

5 | Linear programming with two variables per inequality in poly log time
- Lueker, Megiddo, et al.
(Show Context)
Citation Context ...gramming problems with at most two variables in each constraint and in the objective function were shown to have a strongly polynomial time algorithm by Megiddo [10]. Lueker, Megiddo and Ramachandran =-=[9]-=- gave a polylogarithmic time parallel algorithm for the problem which uses a quasipolynomial number of processors. The best known time bounds for the problem were given in [7, 2]. Cosares, using neste... |

3 |
Strongly polynomial and NC algorithms for detecting cycles in dynamic graphs
- Cohen, Megiddo
(Show Context)
Citation Context ... values of g, rather than the values of the restriction g 2 , at the hyperplanes f1g and f3g. 4. Employing multi-dimensional search The definitions and propositions stated in this section appeared in =-=[3, 5, 2]-=-. They are presented here to allow for an independent reading of this paper. For proofs, the reader is referred to [3, 5, 2]. The multi-dimensional search problem was defined and used in [11] for solv... |

3 |
Strongly polynomial time and NC algorithms for detecting cycles in periodic graphs
- Cohen, Megiddo
- 1990
(Show Context)
Citation Context ..., CA 94305 and IBM Almaden Research Center. Nimrod Megiddo IBM Almaden Research Center, San Jose, CA 95120-6099 and School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel. Abstract In =-=[3, 5, 2]-=- the authors introduced a technique which enabled them to solve the parametric minimum cycle problem with a fixed number of parameters in strongly polynomial time. In the current paper 1 we present th... |

3 |
Using separation algorithms in xed dimensions
- Norton, Plotkin, et al.
- 1992
(Show Context)
Citation Context ...e de ned with respect to the lexicographic order as discussed in the introduction. Below we present some applications of Theorem 8.2. Additional applications were found by Norton, Plotkin, and Tardos =-=[12]-=-. Adding variables to LP's with two variables per inequality. Linear programming problems with at most two variables in each constraint and in the objective function were shown to have a strongly poly... |

2 |
Linear programming in O(n \Theta 3 d ) time
- Clarkson
- 1986
(Show Context)
Citation Context ...ns of maximum and concavity of g with respect to the lexicographic order as follows. We say that a function g : Q ae R d ! R ` is concave with respect to the lexicographic orderslex if for every ff 2 =-=[0; 1]-=- and x; y 2 Q, ffg(x) + (1 \Gamma ff)g(y)slex g(ffx + (1 \Gamma ff)y) : Applications where the range of g is R 2 were given in [6]. In Section 2. we define the problem. In Section 3. we introduce the ... |

2 |
Complexity analysis and algorithms for some flow problems
- Cohen, Megiddo
- 1991
(Show Context)
Citation Context ...ms also has a strongly polynomial time algorithm, and a polylogarithmic time parallel algorithm which uses a quasipolynomial number of processors. Parametric flow problems. Theorem 8.2 was applied in =-=[6]-=- to generate strongly polynomial algorithms for parametric flow problems with a fixed number of param13 eters and to some constrained flow problems with a fixed number of additional constraints. Compl... |

2 |
Maximizing concave functions in xed dimension
- Cohen, Megiddo
- 1990
(Show Context)
Citation Context ...on, Plotkin, and Tardos [12] applied a similar scheme and presented additional applications. Keywords: Complexity, concave-cost network ow, capacitated, global optimization, local optimization. 1 See =-=[4, 2]-=- for an earlier version. 1s1. Introduction A convex optimization problem is a problem of minimizing a convex function g over a convex set S R d .Equivalently, we can consider maximizing a concave func... |

1 |
Complexity analysis and algorithms for some ow problems
- Cohen, Megiddo
- 1991
(Show Context)
Citation Context ...lems also has a strongly polynomial time algorithm, and a polylogarithmic time parallel algorithm which uses a quasipolynomial number of processors. Parametric ow problems. Theorem 8.2 was applied in =-=[6]-=- to generate strongly polynomial algorithms for parametric ow problems with a xed number of param13 ! dseters and to some constrained ow problems with a xed number of additional constraints. Complemen... |