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## Approximating Fractional Multicommodity Flow Independent of the Number of Commodities (1999)

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Citations: | 110 - 8 self |

### Citations

523 | The Geometry of Graphs and Some of its Algorithmic Applications
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Citation Context ...pproximation results for this problem that again round the fractional solution to the LP relaxation of the sparsest cut problem, the most recent algorithms are given by London, Linial, and Rabinovich =-=[22]-=-, and Aumann and Rabani [3]. The sparsest cut problem arises as a subroutine in an algorithm for finding an approximately optimal balanced cut [20]. The balanced cut problem is to partition the vertic... |

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The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1
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Citation Context ...polynomial time using the ellipsoid algorithm, via the equivalence of polynomial time separation and polynomial time optimization for convex polytopes, established by Grötschel, Lovász, and Schrijver =-=[16]-=-. This in turn implies that the primal can also be solve in polynomial time. Given a candidate vector l, the separation algorithm for D is to compute the shortest path for each commodity using l as th... |

325 | Faster and simpler algorithms for multicommodity flow and other fractional packing problems,”
- Garg, Konemann
- 1998
(Show Context)
Citation Context ...umbia University, and while on leave at Center for Operations Research and Econometrics, Louvain-la-Neuve, Belgium. 1s2 L. K. FLEISCHER problem previous best this paper max multiflow O ∗ (ɛ −2 km 2 ) =-=[15, 10]-=- 1 O ∗ (ɛ −2 m 2 ) max concurrent O ∗ (ɛ −2 m(m + k) + k max flows) [10] 1 O ∗ (ɛ −2 m(m + k)) flow O ∗ (ɛ −2 kmn) [19, 24] min cost O ∗ ((ɛ −2 m(m + k) + kmn)I) [10] 1 O ∗ (ɛ −2 m(m + k)I) concurrent... |

261 | Approximation algorithms for fractional packing and covering problems
- Plotkin, Shmoys, et al.
- 1995
(Show Context)
Citation Context ...and Khachiyan [13] describe approximation schemes for block angular linear programs that generalize the uniform capacity maximum and minimum cost concurrent flow problems. Plotkin, Shmoys, and Tardos =-=[23]-=- formulate more general approximation schemes for fractional packing and covering problems, and describe an approximation scheme for the minimum cost concurrent flow with general capacities. They also... |

246 | An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms.
- Leighton, Rao
- 1988
(Show Context)
Citation Context ...gorithms are given by London, Linial, and Rabinovich [22], and Aumann and Rabani [3]. The sparsest cut problem arises as a subroutine in an algorithm for finding an approximately optimal balanced cut =-=[20]-=-. The balanced cut problem is to partition the vertices of a graph into two sets of size at least |V |/3, so that the total capacity of the edges that have one endpoint in each set is minimized. 2. Ma... |

191 | Improved approximation algorithms for the multi-commodity flow problem and local competitive routing in dynamic networks.
- Awerbuch, Leighton
- 1994
(Show Context)
Citation Context ...K. FLEISCHER problem previous best this paper max multiflow O ∗ (ɛ −2 km 2 ) [15, 10] 1 O ∗ (ɛ −2 m 2 ) max concurrent O ∗ (ɛ −2 m(m + k) + k max flows) [10] 1 O ∗ (ɛ −2 m(m + k)) flow O ∗ (ɛ −2 kmn) =-=[19, 24]-=- min cost O ∗ ((ɛ −2 m(m + k) + kmn)I) [10] 1 O ∗ (ɛ −2 m(m + k)I) concurrent flow O ∗ (ɛ −2 kmnI) [14] Fig. 1.1. Comparison of multicommodity flow FPTAS. O ∗ () hides polylog(m). I := log M. the larg... |

165 |
The Maximum Concurrent Flow Problem”,
- Shahrokhi, Matula
- 1990
(Show Context)
Citation Context ... approximation guarantees, it appears that the flow deviation method introduced in their paper yields an approximation guarantee for this problem with only minor adjustments [6]. Shahrokhi and Matula =-=[26]-=- give the first polynomial time, combinatorial algorithm for approximating the maximum concurrent flow problem with uniform capacities, and introduce the use of an exponential length function to model... |

159 | Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications - Garg, Vazirani, et al. - 1996 |

128 | An O(log k) approximate min-cut max-flow theorem and approximation algorithm
- Aumann, Rabani
- 1998
(Show Context)
Citation Context ...s problem that again round the fractional solution to the LP relaxation of the sparsest cut problem, the most recent algorithms are given by London, Linial, and Rabinovich [22], and Aumann and Rabani =-=[3]-=-. The sparsest cut problem arises as a subroutine in an algorithm for finding an approximately optimal balanced cut [20]. The balanced cut problem is to partition the vertices of a graph into two sets... |

109 | The flow deviation method: An approach to store-and-forward communication network design.
- Fratta, Gerla, et al.
- 1973
(Show Context)
Citation Context ...near programming decomposition techniques. Preceeding this work is a 1973 paper by Fratta, Gerla, and Kleinrock that uses very similar techniques for solving minimum cost multicommodity flow problems =-=[8]-=-. While they do not discuss approximation guarantees, it appears that the flow deviation method introduced in their paper yields an approximation guarantee for this problem with only minor adjustments... |

90 | Randomized rounding without solving the linear program
- Young
- 1995
(Show Context)
Citation Context ...tract [10] makes stronger claims, but the actual achievable run times are correctly stated here [9].sAPPROXIMATING FRACTIONAL MULTICOMMODITY FLOW 3 more congested paths to less congested paths. Young =-=[28]-=- describes a randomized algorithm that works by augmenting flow along shortest paths using the exponential length function, instead of augmenting by single commodity minimum cost flows. Shmoys [27] ex... |

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 ...ximating the maximum concurrent flow problem with uniform capacities, and introduce the use of an exponential length function to model the congestion of flow on an edge. Klein, Plotkin, Stein, Tardos =-=[18]-=- improve the complexity of this algorithm using randomization. Leighton, et al. [19] extend [18] to handle graphs with arbitrary capacities, and give improved run times when capacities are uniform. No... |

81 | Cut problems and their applications to divide-and-conquer. In
- Shmoys
- 1997
(Show Context)
Citation Context ...ng [28] describes a randomized algorithm that works by augmenting flow along shortest paths using the exponential length function, instead of augmenting by single commodity minimum cost flows. Shmoys =-=[27]-=- explains how the framework of [23] can be used to approximately solve the maximum multicommodity flow problem. The subproblem he uses is also a shortest path problem. Grigoriadis and Khachiyan [15] r... |

62 |
Fast approximation schemes for convex programs with many blocks and coupling constraints
- Grigoriadis, Khachiyan
- 1994
(Show Context)
Citation Context ...[19] extend [18] to handle graphs with arbitrary capacities, and give improved run times when capacities are uniform. None of these papers considers the versions with costs. Grigoriadis and Khachiyan =-=[13]-=- describe approximation schemes for block angular linear programs that generalize the uniform capacity maximum and minimum cost concurrent flow problems. Plotkin, Shmoys, and Tardos [23] formulate mor... |

53 |
Coordination complexity of parallel price-directive decomposition
- Grigoriadis, Khachiyan
- 1996
(Show Context)
Citation Context ...umbia University, and while on leave at Center for Operations Research and Econometrics, Louvain-la-Neuve, Belgium. 1s2 L. K. FLEISCHER problem previous best this paper max multiflow O ∗ (ɛ −2 km 2 ) =-=[15, 10]-=- 1 O ∗ (ɛ −2 m 2 ) max concurrent O ∗ (ɛ −2 m(m + k) + k max flows) [10] 1 O ∗ (ɛ −2 m(m + k)) flow O ∗ (ɛ −2 kmn) [19, 24] min cost O ∗ ((ɛ −2 m(m + k) + kmn)I) [10] 1 O ∗ (ɛ −2 m(m + k)I) concurrent... |

45 | Adding multiple cost constraints to combinatorial optimization problems, with applications to multicommodity
- Karger, Plotkin
- 1995
(Show Context)
Citation Context ...hm to match the run times of the fastest existing randomized algorithms for the maximum concurrent flow problem. His algorithm uses a “round-robin” approach to routing commodities. Karger and Plotkin =-=[17]-=- use this idea to obtain deterministic algorithms for minimum cost concurrent flow that reduce the dependence of deterministic algorithms on ɛ. For fixed ɛ, their algorithm also improves upon the fast... |

36 | Provably good global routing by a new approximation algorithm for multicommodity flow
- Albrecht
(Show Context)
Citation Context ...presentation may be easier to follow. However, since the publication of an extended abstract of this paper [7], these ideas have been tested and shown to lead to demonstrable improvements in practice =-=[2, 25]-=-. One area of application for obtaining quick approximate solutions to fractional multicommodity flow problems is the field of network design: in VLSI design [2], or in design of telecommunication net... |

32 | An implementation of a combinatorial approximation algorithm for minimum-cost multicommodity flow
- Goldberg, Oldham, et al.
- 1998
(Show Context)
Citation Context ...imate and often exact optimal solutions to block decomposable linear programs by building on the ɛ-approximation methods described in [23, 13]. There has also been earlier experimental work including =-=[12, 14, 21]-=-. In [12], they show that certain aspects of the approximation schemes need to be fine tuned to obtain good performance in practice. For example, one aspect is the choice of step size in each iteratio... |

32 |
Fast Deterministic Approximation for the Multicommodity Flow Problem
- Radzik
- 1995
(Show Context)
Citation Context ...K. FLEISCHER problem previous best this paper max multiflow O ∗ (ɛ −2 km 2 ) [15, 10] 1 O ∗ (ɛ −2 m 2 ) max concurrent O ∗ (ɛ −2 m(m + k) + k max flows) [10] 1 O ∗ (ɛ −2 m(m + k)) flow O ∗ (ɛ −2 kmn) =-=[19, 24]-=- min cost O ∗ ((ɛ −2 m(m + k) + kmn)I) [10] 1 O ∗ (ɛ −2 m(m + k)I) concurrent flow O ∗ (ɛ −2 kmnI) [14] Fig. 1.1. Comparison of multicommodity flow FPTAS. O ∗ () hides polylog(m). I := log M. the larg... |

28 | Approximate minimum-cost multicommodity flows
- Grigoriadis, Khachiyan
- 1996
(Show Context)
Citation Context ... concurrent O ∗ (ɛ −2 m(m + k) + k max flows) [10] 1 O ∗ (ɛ −2 m(m + k)) flow O ∗ (ɛ −2 kmn) [19, 24] min cost O ∗ ((ɛ −2 m(m + k) + kmn)I) [10] 1 O ∗ (ɛ −2 m(m + k)I) concurrent flow O ∗ (ɛ −2 kmnI) =-=[14]-=- Fig. 1.1. Comparison of multicommodity flow FPTAS. O ∗ () hides polylog(m). I := log M. the largest integer M used to specify any of the capacities, costs, and demands. To simplify the run times, we ... |

26 | AND MIHALIS YANNAKAKIS: Approximate maxflow min-(multi)cut theorems and their applications - GARG, VAZIRANI - 1996 |

22 | Implementation of a Combinatorial Multicommodity Flow Algorithm
- Leong, Shor, et al.
- 1993
(Show Context)
Citation Context ...imate and often exact optimal solutions to block decomposable linear programs by building on the ɛ-approximation methods described in [23, 13]. There has also been earlier experimental work including =-=[12, 14, 21]-=-. In [12], they show that certain aspects of the approximation schemes need to be fine tuned to obtain good performance in practice. For example, one aspect is the choice of step size in each iteratio... |

8 | Experiments with a network design algorithm using ffl- approximate linear programs
- Bienstock
- 1996
(Show Context)
Citation Context ...area of application for obtaining quick approximate solutions to fractional multicommodity flow problems is the field of network design: in VLSI design [2], or in design of telecommunication networks =-=[4]-=-. Given pairwise demands, it is desired to build a network with enough capacity to route all demand. The network design problems encountered in practice are typically NP-hard, and difficult to solve. ... |

6 |
Approximation Algorithms for Linear Programming: Theory and Practice.
- Bienstock
- 2001
(Show Context)
Citation Context ... While they do not discuss approximation guarantees, it appears that the flow deviation method introduced in their paper yields an approximation guarantee for this problem with only minor adjustments =-=[6]-=-. Shahrokhi and Matula [26] give the first polynomial time, combinatorial algorithm for approximating the maximum concurrent flow problem with uniform capacities, and introduce the use of an exponenti... |

2 |
Efficient implementation of an approximation algorithm for multicommodity flows
- Sato
- 2000
(Show Context)
Citation Context ...presentation may be easier to follow. However, since the publication of an extended abstract of this paper [7], these ideas have been tested and shown to lead to demonstrable improvements in practice =-=[2, 25]-=-. One area of application for obtaining quick approximate solutions to fractional multicommodity flow problems is the field of network design: in VLSI design [2], or in design of telecommunication net... |

1 |
An implementation of the exponential potential reduction method for general linear programs. Working paper
- Bienstock
- 1999
(Show Context)
Citation Context ...lgorithms that deliver solutions that are provably close to optimal. Experimental results to date suggest that these techniques can lead to significantly faster solution times. For example, Bienstock =-=[5]-=- has reported significant speedups in obtaining approximate and often exact optimal solutions to block decomposable linear programs by building on the ɛ-approximation methods described in [23, 13]. Th... |

1 | Talk at Oberwolfach - Bienstock - 1999 |