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

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Citations: | 325 - 5 self |

### Citations

696 | Network Flows - Ahuja, Magnanti, et al. - 1993 |

519 | Smooth minimization of non-smooth functions - Nesterov |

261 | 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... |

191 | Improved approximation algorithms for the multi-commodity flow problem and local competitive routing in dynamic networks.
- Awerbuch, Leighton
- 1994
(Show Context)
Citation Context ...e 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. Plotkin, Shmoys and Tardos [11] and Grigoriadis and Khachiyan [7] observed that a similar techniq... |

165 |
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... |

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

110 | Approximating fractional multicommodity flow independent of the number of commodities - Fleischer |

109 | Divide-and-conquer approximation algorithms via spreading metrics
- Even, Naor, et al.
(Show Context)
Citation Context ...r js where distssfP is the distance from to under the length function P and js is a function only of the size of . For the linear arrangement problem js *s, , : s, , :seds=-=[2]-=- while for the problem of computing a -separator js is defined as , , : , E , [3]. Since the length function P is positive, the shortest paths from to the other vertices in forms a tre... |

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... |

88 | 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... |

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

62 |
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... |

60 | 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 price-directive decomposition - Grigoriadis, Khachiyan - 1996 |

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

37 |
Speeding up linear programming 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
- 1995
(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... |

29 | Faster Approximation Schemes for Fractional Multi-commodity Flow - Karakostas - 2002 |

28 | 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... |

24 | Fast and simple approximation schemes for generalized flow, - Fleischer, Wayne - 2002 |

21 |
An exponential-Function Reduction Method for BlockAngular Convex Programs.”
- Grigoriadis, Kahchiyan
- 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 ... |

21 | Approximation algorithms for multicommodity flow and shop scheduling problems. - Stein - 1992 |

16 |
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... |

6 |
Approximation Algorithms for NP-Hard Problems, chapter Cut problems and their application to divide-and-conquer. PWS series in computer science
- 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 ... |

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

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 ... |