#### DMCA

## Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs (2010)

### Cached

### Download Links

Citations: | 40 - 5 self |

### Citations

1558 |
Modern Graph Theory,
- Bollobas
- 1998
(Show Context)
Citation Context ...s about electrical flows in networks of resistors and present a theorem that allows us to quickly approximate these flows. For an in-depth treatment of the background material, we refer the reader to =-=[6]-=-. We begin by assigning a resistance re > 0 to each edge e ∈ E, and we collect these resistances into a vector r ∈ IR m . For a given s-t flow f, we define its energy (with respect to r) as Er (f) := ... |

698 |
Network flows,
- Ahuja, Magnanti, et al.
- 1993
(Show Context)
Citation Context ...tant c.1 Introduction The maximum s-t flow problem and its dual, the minimum s-t cut problem, are two of the most fundamental and extensively studied problems in Operations Research and Optimization =-=[18, 1]-=-. They have many applications (see [2]) and are often used as subroutines in other algorithms (see [3, 19]). Many advances have been made in the development of algorithms for this problem (see Goldber... |

434 |
Maximal flow through a network,”
- Ford, Fulkerson
- 1956
(Show Context)
Citation Context ... simple scaling, we can assume that all edge capacities are integers between 1 and 2m2 /ɛ. The minimum s-t cut problem is that of finding an s-t cut of minimum capacity. The Max Flow-Min Cut Theorem (=-=[11, 8]-=-) states that the capacity of the minimum s-t cut is equal to F ∗ , the value of the maximum s-t flow. In particular, the Max Flow-Min Cut Theorem implies that one can use the capacity of any s-t cut ... |

325 | Faster and simpler algorithms for multicommodity flow and other fractional packing problems,”
- Garg, Konemann
- 1998
(Show Context)
Citation Context ...y modifying the one described here. The algorithm will employ the multiplicative weights update method, a framework established by Arora, Hazan and Kale [3] to encompass proof techniques exploited in =-=[17, 21, 10, 12]-=-. In our setting, one can understand the multiplicative weights method as a way of taking an algorithm that solves a flow problem very crudely and, by calling it repeatedly, converts it into an algori... |

260 | Fast approximation algorithms for fractional packing and covering problems.
- Plotkin, Shmoys, et al.
- 1995
(Show Context)
Citation Context ...y modifying the one described here. The algorithm will employ the multiplicative weights update method, a framework established by Arora, Hazan and Kale [3] to encompass proof techniques exploited in =-=[17, 21, 10, 12]-=-. In our setting, one can understand the multiplicative weights method as a way of taking an algorithm that solves a flow problem very crudely and, by calling it repeatedly, converts it into an algori... |

167 |
Maximal Flow Through a Network,” Canadian
- Ford, Fulkerson
- 1956
(Show Context)
Citation Context ...simple scaling, we can assume that all edge capacities are integers between 1 and 2m2 /ɛ. The minimum s-t cut problem is that of finding the s-t cut of minimum capacity. The Max Flow-Min Cut Theorem (=-=[10, 8]-=-) states that the capacity of the minimum s-t cut is equal to F ∗ , the value of the maximum s-t flow. 3In particular, the Max Flow-Min Cut Theorem implies that one can use the capacity of any s-t cu... |

150 |
Beyond the flow decomposition barrier.
- Goldberg, Rao
- 1998
(Show Context)
Citation Context ...e many applications (see [2]) and are often used as subroutines in other algorithms (see [3, 19]). Many advances have been made in the development of algorithms for this problem (see Goldberg and Rao =-=[13]-=- for an overview). However, for the basic problem of computing or (1 − ɛ)approximating the maximum flow in undirected, unit-capacity graphs with m = O(n) edges, the asymptotically fastest known algori... |

147 | The multiplicative weights update method: a meta algorithm and applications.
- Arora, Hazan, et al.
- 2005
(Show Context)
Citation Context ...the most fundamental and extensively studied problems in Operations Research and Optimization [18, 1]. They have many applications (see [2]) and are often used as subroutines in other algorithms (see =-=[3, 19]-=-). Many advances have been made in the development of algorithms for this problem (see Goldberg and Rao [13] for an overview). However, for the basic problem of computing or (1 − ɛ)approximating the m... |

113 |
Network flow and testing graph connectivity.
- Even, Tarjan
- 1975
(Show Context)
Citation Context ... computing or (1 − ɛ)approximating the maximum flow in undirected, unit-capacity graphs with m = O(n) edges, the asymptotically fastest known algorithm is the one developed in 1975 by Even and Tarjan =-=[9]-=-, which takes time O(n 3/2 ). Despite 35 years of extensive work on the problem, this bound has not been improved. In this paper, we introduce a new approach to computing approximately maximum s-t flo... |

110 | Approximating fractional multicommodity flow independent of the number of commodities
- Fleischer
(Show Context)
Citation Context ...y modifying the one described here. The algorithm will employ the multiplicative weights update method, a framework established by Arora, Hazan and Kale [3] to encompass proof techniques exploited in =-=[17, 21, 10, 12]-=-. In our setting, one can understand the multiplicative weights method as a way of taking an algorithm that solves a flow problem very crudely and, by calling it repeatedly, converts it into an algori... |

106 |
A note on the maximum flow through a network,”
- 25Elias, Feinstein, et al.
- 1956
(Show Context)
Citation Context ... simple scaling, we can assume that all edge capacities are integers between 1 and 2m2 /ɛ. The minimum s-t cut problem is that of finding an s-t cut of minimum capacity. The Max Flow-Min Cut Theorem (=-=[11, 8]-=-) states that the capacity of the minimum s-t cut is equal to F ∗ , the value of the maximum s-t flow. In particular, the Max Flow-Min Cut Theorem implies that one can use the capacity of any s-t cut ... |

90 | Randomized rounding without solving the linear program
- Young
- 1995
(Show Context)
Citation Context |

75 |
Nearlylinear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems’,
- Spielman, Teng
- 2014
(Show Context)
Citation Context ...resistor networks. An approximate solution to each electrical flow problem can be found in time Õ (m) using recently developed algorithms for solving systems of linear equations in Laplacian matrices =-=[15, 20]-=-. There is a simple physical intuition that underlies our approach, which we describe here in the case of a graph with unit edge capacities. We begin by thinking of each edge of the input graph as a r... |

52 |
Approximating s-t minimum cuts in O(n2) time.
- Benczúr, Karger
- 1996
(Show Context)
Citation Context ...h the graph smoothing and sampling techniques of Karger [14], we can get a (1 − ɛ)-approximately maximum s-t flow in time Õ ( mn1/3ɛ−11/3) . Furthermore, by applying the cut algorithm to a sparsifier =-=[4]-=- of the input graph, we can compute a (1 + ɛ)-approximately minimum s-t cut in time Õ ( m + n 4/3 ɛ −16/3) . We remark that the results in this paper immediately improve the running time of algorithms... |

29 | Breaking the multicommodity flow barrier for o( √ log n)- approximations to sparsest cut
- Sherman
- 2009
(Show Context)
Citation Context ...the most fundamental and extensively studied problems in Operations Research and Optimization [18, 1]. They have many applications (see [2]) and are often used as subroutines in other algorithms (see =-=[3, 19]-=-). Many advances have been made in the development of algorithms for this problem (see Goldberg and Rao [13] for an overview). However, for the basic problem of computing or (1 − ɛ)approximating the m... |

28 |
Faster approximate lossy generalized flow via interior point algorithms.
- Daitch, Spielman
- 2008
(Show Context)
Citation Context ...refer the reader to the the paper of Goldberg and Rao for a survey of previous breakthroughs in the development of algorithms for computing maximum s-t flows. In more recent work, Daitch and Spielman =-=[7]-=- showed that fast solvers for Laplacian linear systems [20, 15] could be used to make interior-point algorithms for the maximum flow and minimumcost flow problems run in time Õ ( m 3/2 log U ) , and M... |

25 |
Approaching optimality for solving SDD systems
- Koutis, Miller, et al.
- 2010
(Show Context)
Citation Context ...resistor networks. An approximate solution to each electrical flow problem can be found in time Õ (m) using recently developed algorithms for solving systems of linear equations in Laplacian matrices =-=[15, 20]-=-. There is a simple physical intuition that underlies our approach, which we describe here in the case of a graph with unit edge capacities. We begin by thinking of each edge of the input graph as a r... |

22 | Applications of network optimization
- Ahuja, Magnanti, et al.
- 1995
(Show Context)
Citation Context ...w problem and its dual, the minimum s-t cut problem, are two of the most fundamental and extensively studied problems in Operations Research and Optimization [18, 1]. They have many applications (see =-=[2]-=-) and are often used as subroutines in other algorithms (see [3, 19]). Many advances have been made in the development of algorithms for this problem (see Goldberg and Rao [13] for an overview). Howev... |

19 | Fast Approximation Algorithms for Cut-Based Problems in Undirected Graphs.
- Madry
- 2010
(Show Context)
Citation Context ...ed that fast solvers for Laplacian linear systems [20, 15] could be used to make interior-point algorithms for the maximum flow and minimumcost flow problems run in time Õ ( m 3/2 log U ) , and Mądry =-=[16]-=- showed that one can approximate a wide range of cut problems, including the minimum s-t cut problem, within a polylogarithmic factor in almost linear time. 1.2 Outline We begin the technical part of ... |

17 | Randomized approximation schemes for cuts and flows in capacitated graphs
- Benczúr, Karger
(Show Context)
Citation Context ...Previous Work on Maximum Flows and Minimum Cuts The best previously known algorithms for the problems studied here are obtained by combining techniques of Goldberg and Rao [13] and Benczúr and Karger =-=[5]-=-. In a breakthrough paper, Goldberg and Rao [13] developed an algorithm for computing exact maximum s-t flows in directed or undirected capacitated graphs in time O(m min(n 2/3 , m 1/2 ) log(n 2 /m) l... |

12 | Better random sampling algorithms for flows in undirected graphs
- Karger
- 1998
(Show Context)
Citation Context ...t from the vertex potentials 2 . This will give us algorithms for both problems that run in time Õ ( m 4/3 · poly(1/ɛ) ) . By combining this with the graph smoothing and sampling techniques of Karger =-=[14]-=-, we can get a (1 − ɛ)-approximately maximum s-t flow in time Õ ( mn1/3ɛ−11/3) . Furthermore, by applying the cut algorithm to a sparsifier [4] of the input graph, we can compute a (1 + ɛ)-approximate... |

2 |
Combinatorial Optimization, Volume A. Number 24 in Algorithms and Combinatorics
- Schrijver
- 2003
(Show Context)
Citation Context ...tant c.1 Introduction The maximum s-t flow problem and its dual, the minimum s-t cut problem, are two of the most fundamental and extensively studied problems in Operations Research and Optimization =-=[18, 1]-=-. They have many applications (see [2]) and are often used as subroutines in other algorithms (see [3, 19]). Many advances have been made in the development of algorithms for this problem (see Goldber... |