## Faster and simpler algorithms for multicommodity flow and other fractional packing problems (1998)

### Cached

### Download Links

- [www.csc.liv.ac.uk]
- [www.math.uwaterloo.ca]
- [cgi.csc.liv.ac.uk]
- [www.cse.iitd.ernet.in]
- [www.andrew.cmu.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | In Proceedings of the 39th Annual Symposium on Foundations of Computer Science |

Citations: | 279 - 5 self |

### BibTeX

@INPROCEEDINGS{Garg98fasterand,

author = {Naveen Garg and Jochen Könemann},

title = {Faster and simpler algorithms for multicommodity flow and other fractional packing problems},

booktitle = {In Proceedings of the 39th Annual Symposium on Foundations of Computer Science},

year = {1998},

pages = {300--309},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We provide a different approach to these problems which yields faster and much simpler algorithms. Our approach also allows us to substitute shortest path computations for min-cost flow computations in computing maximum concurrent flow and min-cost multicommodity flow; this yields much faster algorithms when the number of commodities is large.

### Citations

600 | Network flows - Ahuja, Magnanti, et al. - 1993 |

279 | 2005b. Smooth minimization of non-smooth functions - Nesterov |

238 | Fast approximation algorithms for fractional packing and covering problems
- Plotkin, Shmoys, et al.
- 1995
(Show Context)
Citation Context ...fined to the case of arbitrary edge capacities by Leighton et.al. [10], Goldberg [4] and Radzik [12] to obtain better running times; see Table 1 for the current best bound. Plotkin, Shmoys and Tardos =-=[11]-=- and Grigoriadis and Khachiyan [7] observed that a similar technique could be applied to solve any fractional packing or covering problem. Their approach, for packing problems, starts with an infeasib... |

175 | Fast approximation algorithms for multicommodity flow problems - Leighton, Makedon, et al. - 1995 |

153 |
The Maximum Concurrent Flow Problem
- Shahrokhi, Matula
- 1990
(Show Context)
Citation Context ...le flow which is almost maximum. Note that the length of an edge at any step is exponential in the total flow going through the edge. Such a length function was first proposed by Shahrokhi and Matula =-=[13]-=- who Supported by the EU ESPRIT LTR Project N. 20244 (ALCOM-IT). Work done while the author was at the Max-Planck-Institut fur Informatik, Im Stadtwald, 66123 Saarbrucken, Germany. y Work done while t... |

123 | Potential function methods for approximately solving linear programming problems: theory and practice - BIENSTOCK - 2001 |

103 | Divide-and-conquer Approximation Algorithms via Spreading Metrics - Even, Naor, et al. - 1995 |

95 | Approximating Fractional Multicommodity Flow Independent of the Number of Commodities - Fleischer |

90 | Randomized rounding without solving the linear program
- Young
- 1995
(Show Context)
Citation Context ... only pseudo-polynomial, [11] suggest different ways of reducing the width of the problem. In a significant departure from this line of research and motivated by ideas from randomized rounding, Young =-=[17]-=- proposed an oblivious rounding approach to packing problems. Young's approach has the essential ingredient of previous approaches --- a length function which measures, and is exponential in, the exte... |

84 | Faster Approximation Algorithms for the Unit Capacity Concurrent Flow Problem with Applications to Routing and Finding Sparse Cuts
- Klein, Plotkin, et al.
- 1994
(Show Context)
Citation Context ...arding edge capacities and then to reroute this, iteratively, along short paths so as to reduce the maximum congestion on any edge. The running time of [13] was improved significantly by Klein et.al. =-=[9]-=-. It was then extended and refined to the case of arbitrary edge capacities by Leighton et.al. [10], Goldberg [4] and Radzik [12] to obtain better running times; see Table 1 for the current best bound... |

60 |
Fast approximation schemes for convex programs with many blocks and coupling constraints
- Grigoriadis, Khachiyan
- 1994
(Show Context)
Citation Context ... capacities by Leighton et.al. [10], Goldberg [4] and Radzik [12] to obtain better running times; see Table 1 for the current best bound. Plotkin, Shmoys and Tardos [11] and Grigoriadis and Khachiyan =-=[7]-=- observed that a similar technique could be applied to solve any fractional packing or covering problem. Their approach, for packing problems, starts with an infeasible solution. The amount by which a... |

57 | Fast approximate graph partitioning algorithms
- Even, Naor, et al.
- 1999
(Show Context)
Citation Context ... Problem Previous Best Our running time Improvement Max. multicomm. O(! \Gamma3 m 2 log m) [14] mC 1 T sp ! \Gamma1 flow Fractional Packing [5] mC 1 T orc Spreading metrics O(! \Gamma3 nm log nT sp ) =-=[3]-=- mC 1 (nT sp ) ! \Gamma1 Maximum O(k(! \Gamma2 + log k) log nT mcf ) (2k log k)C 2 T mcf In constants concurrent flow [12, 10] (2k log k +m)C 2 T sp For ksm=n Max. cost-bounded O(k(! \Gamma2 + log k) ... |

53 | Coordination complexity of parallel pricedirective decomposition - Grigoriadis, Khachiyan - 1996 |

47 | Adding multiple cost constraints to combinatorial optimization problems, with applications to multicommodity flows - Karger, Plotkin - 1995 |

46 | Sequential and parallel algorithms for mixed packing and covering - Young - 2001 |

35 |
Speeding up Linear Progra,mming Using Fast Matrix Multiplication
- Vaidya
- 1989
(Show Context)
Citation Context ...iven multicommodity flow instance. While this problem (and all other problems considered in this paper) can be formulated as a linear program and solved to optimality using fast matrix multiplication =-=[16]-=-, in [13] were mainly interested in providing fast, possibly approximate, combinatorial algorithms. Their procedure, which applied only to the case of uniform edge capacities, computed a (1 + !)-appro... |

32 |
Fast deterministic approximation for the multicommodity flow problem
- Radzik
- 1997
(Show Context)
Citation Context ...ge. The running time of [13] was improved significantly by Klein et.al. [9]. It was then extended and refined to the case of arbitrary edge capacities by Leighton et.al. [10], Goldberg [4] and Radzik =-=[12]-=- to obtain better running times; see Table 1 for the current best bound. Plotkin, Shmoys and Tardos [11] and Grigoriadis and Khachiyan [7] observed that a similar technique could be applied to solve a... |

26 | Approximate minimum-cost multicommodity flows
- Grigoriadis, Khachiyan
- 1996
(Show Context)
Citation Context ... In constants concurrent flow [12, 10] (2k log k +m)C 2 T sp For ksm=n Max. cost-bounded O(k(! \Gamma2 + log k) log n log(! \Gamma1 k) (2k log k + 1)C 2 T mcf log(! \Gamma1 k) concurrent flow T mcf ) =-=[6]-=- (2k log k +m+ 1)C 2 T sp For ksm=n Table 1. A summary of our results Consider the length function l i \Gamma l 0 . Note that D(l i \Gamma l 0 ) = D(i) \Gamma D(0) and ff(l i \Gamma l 0 )sff(i) \Gamma... |

20 |
An exponential-function reduction method for block-angular convex programs
- Grigoriadis, Khachiyan
- 1995
(Show Context)
Citation Context ... and straightforward strongly-polynomial combinatorial approximation algorithm for the fractional packing problem (Section 3). The earlier algorithm for this problem, due to Grigoriadis and Khachiyan =-=[5]-=- reduced the problem to two resource sharing problems. Our approach yields a new, very natural, algorithm for maximum concurrent flow (Section 5) which extends in a straightforward manner to min-cost ... |

20 | Faster approximation schemes for fractional multicommodity flow problems - Karakostas - 2002 |

19 | Approximation algorithms for multicommodity flow and scheduling problems - Stein - 1992 |

18 | Fast and simple approximation schemes for generalized flow - Fleischer, Wayne - 1999 |

15 |
A natural randomization strategy for multicommodity flow and related algorithms
- Goldberg
- 1992
(Show Context)
Citation Context ...stion on any edge. The running time of [13] was improved significantly by Klein et.al. [9]. It was then extended and refined to the case of arbitrary edge capacities by Leighton et.al. [10], Goldberg =-=[4]-=- and Radzik [12] to obtain better running times; see Table 1 for the current best bound. Plotkin, Shmoys and Tardos [11] and Grigoriadis and Khachiyan [7] observed that a similar technique could be ap... |

7 |
Divideand -conquer approximation algorithms via spreading metrics
- Even, Naor, et al.
- 1995
(Show Context)
Citation Context ...)sf(S) where dist r;v (l) is the distance from r to v under the length function l and f() is a function only of the size of S. For the linear arrangement problem f(S) = (jSj \Gamma 1)(jSj \Gamma 3)=4 =-=[2]-=- while for the problem of computing a ae-separator 1 f(S) is defined as jSj \Gamma aejV j [3]. Since the length function l is positive, the shortest paths from r to the other vertices in S forms a tre... |

5 |
Approximation Algorithms for NP-hard Problems, chapter Approximation algorithms for Cut problems and their application to divide-and-conquer, pages 192–235. PWS
- Shmoys
- 1997
(Show Context)
Citation Context ...a1 )ff(i \Gamma 1) which implies that D(i) = D(0) + ffl i X j=1 (f j \Gamma f j \Gamma1 )ff(j \Gamma 1) (1) 2 Problem Previous Best Our running time Improvement Max. multicomm. O(! \Gamma3 m 2 log m) =-=[14]-=- mC 1 T sp ! \Gamma1 flow Fractional Packing [5] mC 1 T orc Spreading metrics O(! \Gamma3 nm log nT sp ) [3] mC 1 (nT sp ) ! \Gamma1 Maximum O(k(! \Gamma2 + log k) log nT mcf ) (2k log k)C 2 T mcf In ... |

1 |
An exponential function reduction method for block angular convex programs
- Grigoriadis, Khachiyan
- 1993
(Show Context)
Citation Context ... and straightforward strongly-polynomial combinatorial approximation algorithm for the fractional packing problem (Section 3). The earlier algorithm for this problem, due to Grigoriadis and Khachiyan =-=[5]-=- reduced the problem to two resource sharing problems. Our approach yields a new, very natural, algorithm for maximum concurrent flow (Section 5) which extends in a straightforward manner to min-cost ... |

1 | Approximating fractional packings and coverings in O(1/epsilon) iterations - Bienstock, Iyengar |