## Fast and simple approximation schemes for generalized flow (2001)

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Venue: | Math. Program., Ser. A |

Citations: | 17 - 3 self |

### BibTeX

@MISC{Fleischer01fastand,

author = {Lisa K. Fleischer and Kevin D. Wayne},

title = { Fast and simple approximation schemes for generalized flow},

year = {2001}

}

### Years of Citing Articles

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### Abstract

We present fast and simple fully...

### Citations

1436 | A note on two problems in connexion with graphs - Dijkstra - 1959 |

1408 |
Network Flows: Theory, Algorithms and Applications
- Ahuja, Magnanti, et al.
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Citation Context ...ssed materials into finished products, acres into feed into fattened cattle, crude oil into processed oil, and machine time into completed orders. For more information and examples, see Ahuja, et al. =-=[3]-=- or Glover, et al. [12]. In this paper, we design fast and simple approximation schemes for all of these problems. Our goal is to find an ffl-approximate solution for any error parameter ffl ? 0. For ... |

811 |
Programming and Extensions
- Dantzig, Linear
- 1963
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Citation Context ...n factors, and demands. To simplify the run times, we use Õ(f)todenotef logO(1) m. 1.1 Previous work Generalized flow has a rich history. The problem was first studied by Kantorovich [21] and Dantzig=-= [8]-=-. All of our problems can be solved exactly via general purpose linear programming techniques, including simplex, ellipsoid, and interior point methods. Researchers have also designed efficient combin... |

765 | FZows in Networks - FORD, FULKERSON - 1962 |

575 | Fibonacci heaps and their uses in improved network optimization algorithms - Fredman, Tarjan - 1984 |

324 | On a routing problem - Bellman - 1958 |

269 | Faster and simpler algorithms for multicommodity flow and other fractional packing problems
- Garg, K¨onemann
- 1998
(Show Context)
Citation Context ...yan [17] extended the method further to solve more general fractional packing and covering problems. Goldberg [13] proposed a faster randomized version; Radzik [27] derandomized it. Garg and Könemann=-= [11] s-=-implified the method for packing problems, drawing on ideas from Young [38]. Very recently, Oldham [25] proposed FPTAS’s for a variety of generalized flow problems, using the fractional packing fram... |

232 | Applying parallel computation algorithms in the design of serial algorithms
- Megiddo
- 1983
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Citation Context ...m [25] proposed an algorithm for directly solving the generalized shortest path problem that matches the Õ(mn2 ) complexity bound. His algorithm combines the Aspvall-Shiloach procedure with Megiddo��=-=�s [24]-=- parametric search. 2.3.2 Generalized shortest paths in lossy networks In the case when there are no flow generating cycles, the optimality conditions described in the previous section imply that the ... |

232 | Fast approximation algorithms for fractional packing and covering problems," extended abstract
- Plotkin, Shmoys, et al.
(Show Context)
Citation Context ...paths with respect to the exponential length function. The method was refined by Klein et al. [22] and extended to handle arbitrary arc capacities by Leighton et al. [23]. Plotkin, Shmoys, and Tardos =-=[26]-=- and Grigoriadis and Khachiyan [17] extended the method further to solve more general fractional packing and covering problems. Goldberg [13] proposed a faster randomized version; Radzik [27] derandom... |

173 | Fast approximation algorithms for multicommodity flow problems
- Leighton, Makedon, et al.
- 1995
(Show Context)
Citation Context ...atively route flow along shortest paths with respect to the exponential length function. The method was refined by Klein et al. [22] and extended to handle arbitrary arc capacities by Leighton et al. =-=[23]-=-. Plotkin, Shmoys, and Tardos [26] and Grigoriadis and Khachiyan [17] extended the method further to solve more general fractional packing and covering problems. Goldberg [13] proposed a faster random... |

162 |
Approximation schemes for the restricted shortest path problem
- Hassin
- 1992
(Show Context)
Citation Context ...+ ɛ). Now, the scaled flow has value at least (1 − ɛ)/(1 + ɛ) ≥ 1 − 2ɛ and has cost at most B ∗ . To find a suitable approximation to B ∗ , we use the geometric-mean binary search techni=-=que of Hassin [18] (§4-=-). Given a lower bound LB and an upper bound UB on the desired value B ∗ ,conventional binary search uses the arithmetic mean (LB+UB)/2 and shrinks the difference UB−LB in half. Our goal is actual... |

150 |
The maximum concurrent flow problem
- Shahrokhi, Matula
- 1990
(Show Context)
Citation Context ...nomial combinatorial algorithms for generalized multicommodity flow. Our approximation schemes build upon combinatorial approximation schemes for traditional multicommodity flow. Shahrokhi and Matula =-=[30]-=- proposed a FPTAS for the maximum concurrent flow problem with uniform arc capacities. They introduced a length function which is exponential in the total flow going through that arc. They iteratively... |

98 | The shortest path through a maze - Moore - 1959 |

95 | Approximating fractional multicommodity flow independent of the number of commodities
- Fleischer
- 2000
(Show Context)
Citation Context ...-generating cycles. In Theorem 4.2, this subroutine is Procedure 2.4 in conjunction with Megiddo’s parametric search [24]. This is essentially Oldham’s [25] algorithm. Adapting and extending ideas=-= in [9]-=-, we propose a simpler alternative to parametric search, and improve the run time by a factor of n. The first fact we use is that, in the packing framework, it is not necessary to find the most violat... |

88 | 1995]: Randomized Rounding Without Solving the Linear Program
- Young
- 1995
(Show Context)
Citation Context ... and covering problems. Goldberg [13] proposed a faster randomized version; Radzik [27] derandomized it. Garg and Könemann [11] simplified the method for packing problems, drawing on ideas from Young=-= [38]. V-=-ery recently, Oldham [25] proposed FPTAS’s for a variety of generalized flow problems, using the fractional packing framework of Garg and Könemann [11]. When this framework is applied to traditiona... |

84 | Faster approximation algorithms for the unit capacity concurrent problem with applications to routing and sparse cuts
- Klein, Plotkin, et al.
- 1994
(Show Context)
Citation Context ...which is exponential in the total flow going through that arc. They iteratively route flow along shortest paths with respect to the exponential length function. The method was refined by Klein et al. =-=[22]-=- and extended to handle arbitrary arc capacities by Leighton et al. [23]. Plotkin, Shmoys, and Tardos [26] and Grigoriadis and Khachiyan [17] extended the method further to solve more general fraction... |

71 |
Deciding linear inequalities by computing loop residues
- Shostak
- 1981
(Show Context)
Citation Context ...igger than, less than, or equal to a trial value L. Aspvall and Shiloach [4] give a O(mn) time Bellman-Ford style algorithm for this procedure. Their algorithm exploits structure described by Shostak =-=[31]. Proced-=-ure 2.4 Let L ∗ denote the value of the generalized shortest path. Given L, determine whether L = L ∗ , L<L ∗ ,orL>L ∗ . Lemma 2.5 (Aspvall and Shiloach [4]) There exists a O(mn) time algorith... |

59 |
Fast approximation schemes for convex programs with many blocks and coupling constraints
- Grigoriadis, Khachiyan
- 1994
(Show Context)
Citation Context ...al length function. The method was refined by Klein et al. [22] and extended to handle arbitrary arc capacities by Leighton et al. [23]. Plotkin, Shmoys, and Tardos [26] and Grigoriadis and Khachiyan =-=[17] -=-extended the method further to solve more general fractional packing and covering problems. Goldberg [13] proposed a faster randomized version; Radzik [27] derandomized it. Garg and Könemann [11] sim... |

55 |
Simple and fast algorithms for linear and integer programs with two variables per inequality
- Hochbaum, Naor
- 1994
(Show Context)
Citation Context ...st right. The presence of gain factors makes computing shortest paths more complicated and expensive than standard Bellman-Ford. Currently, the best complexity bound for the problem is Õ(mn2 ) due to=-= [6, 19, 25]. 1-=-.2 Our contributions We refine the generalized flow FPTAS’s of Oldham [25]. The crucial subroutine in [25] is a generalized shortest path computation, which requires Õ(mn2 ) time using any of the s... |

50 | Coordination complexity of parallel price-directive decomposition - Grigoriadis, Khachiyan - 1996 |

40 |
A polynomial time algorithm for solving systems of linear inequalities with two variables per inequality
- Aspvall, Shiloach
- 1980
(Show Context)
Citation Context ...broutine to solve a version of this shortest path problem with gain factors. All efficient combinatorial methods for this subroutine make use of a Bellman-Ford style procedure of Aspvall and Shiloach =-=[4]-=- that tests whether or not some guess on the generalized shortest path value is too big, too small, or just right. The presence of gain factors makes computing shortest paths more complicated and expe... |

38 |
Mathematical methods of organizing and planning production
- Kantorovich
- 1960
(Show Context)
Citation Context ...ities, costs, gain factors, and demands. To simplify the run times, we use Õ(f)todenotef logO(1) m. 1.1 Previous work Generalized flow has a rich history. The problem was first studied by Kantorovich=-= [21]-=- and Dantzig [8]. All of our problems can be solved exactly via general purpose linear programming techniques, including simplex, ellipsoid, and interior point methods. Researchers have also designed ... |

34 |
Speeding up linear programming using fast matrix multiplication
- Vaidya
- 1989
(Show Context)
Citation Context ...r run times for each problem and compares them with previous work. 3sExact algorithm for generalized flow Maximum flow Previous best FPTAS Our FPTAS Õ(m3I)[16] Õ(log ɛ−1m2n) [28] Õ(ɛ−2m2 ) Õ=-=(m1.5n2I) [34] Õ(log ɛ−1 Minimum cost flo-=-w m(m + n log I)) [29, 32] Õ(m1.5n2I) [34] Õ(log ɛ−1m2n2 )[36] Õ(ɛ−2m2J) † Õ(ɛ−2m2 Maximum multicommodity flow nJ) Õ(k2.5m1.5n2I)[34] Õ(ɛ−2km2n2 ) [25] Õ(ɛ−2m2 ) † Õ((k0.5m3... |

32 |
Fast deterministic approximation for the multicommodity flow problem
- Radzik
- 1995
(Show Context)
Citation Context ...d Tardos [26] and Grigoriadis and Khachiyan [17] extended the method further to solve more general fractional packing and covering problems. Goldberg [13] proposed a faster randomized version; Radzik =-=[27] de-=-randomized it. Garg and Könemann [11] simplified the method for packing problems, drawing on ideas from Young [38]. Very recently, Oldham [25] proposed FPTAS’s for a variety of generalized flow pro... |

27 |
Improved algorithms for linear inequalities with two variables per inequality
- Cohen, Megiddo
- 1994
(Show Context)
Citation Context ...st right. The presence of gain factors makes computing shortest paths more complicated and expensive than standard Bellman-Ford. Currently, the best complexity bound for the problem is Õ(mn2 ) due to=-= [6, 19, 25]. 1-=-.2 Our contributions We refine the generalized flow FPTAS’s of Oldham [25]. The crucial subroutine in [25] is a generalized shortest path computation, which requires Õ(mn2 ) time using any of the s... |

26 | Combinatorial Algorithms for the Generalized Circulation Problem
- Goldberg, Plotkin, et al.
- 1988
(Show Context)
Citation Context ...lipsoid, and interior point methods. Researchers have also designed efficient combinatorial algorithms that exploit the underlying network flow structure of the problem. Goldberg, Plotkin, and Tardos =-=[14]-=- designed the first polynomial-time combinatorial algorithms for generalized maximum flow. Their algorithms were refined and improved upon in [15, 16, 29] with the fastest algorithm developed so far b... |

19 | Generalized Maximum Flow Algorithms
- Wayne
- 1999
(Show Context)
Citation Context ...improved upon in [15, 16, 29] with the fastest algorithm developed so far by Goldfarb, Jin, and Orlin [16]. For generalized maximum flow, researchers have also developed fast approximation schemes in =-=[7, 28, 32]-=-. Very recently, Wayne [36] proposed the first polynomial combinatorial algorithms for generalized minimum cost flow. There are no known exact polynomial combinatorial algorithms for generalized multi... |

16 | A polynomial combinatorial algorithm for generalized minimum cost flow
- Wayne
- 1999
(Show Context)
Citation Context ... the fastest algorithm developed so far by Goldfarb, Jin, and Orlin [16]. For generalized maximum flow, researchers have also developed fast approximation schemes in [7, 28, 32]. Very recently, Wayne =-=[36]-=- proposed the first polynomial combinatorial algorithms for generalized minimum cost flow. There are no known exact polynomial combinatorial algorithms for generalized multicommodity flow. Our approxi... |

15 |
A natural randomization strategy for multicommodity flow and related algorithms
- Goldberg
- 1992
(Show Context)
Citation Context ...ities by Leighton et al. [23]. Plotkin, Shmoys, and Tardos [26] and Grigoriadis and Khachiyan [17] extended the method further to solve more general fractional packing and covering problems. Goldberg =-=[13] -=-proposed a faster randomized version; Radzik [27] derandomized it. Garg and Könemann [11] simplified the method for packing problems, drawing on ideas from Young [38]. Very recently, Oldham [25] prop... |

13 | Improved Interior Point Algorithms for Exact and Approximate Solutions of Multicommodity Flow Problems
- Kamath, Palmon
- 1994
(Show Context)
Citation Context ... flow m(m + n log I)) [29, 32] Õ(m1.5n2I) [34] Õ(log ɛ−1m2n2 )[36] Õ(ɛ−2m2J) † Õ(ɛ−2m2 Maximum multicommodity flow nJ) Õ(k2.5m1.5n2I)[34] Õ(ɛ−2km2n2 ) [25] Õ(ɛ−2m2 ) † Õ((k0=-=.5m3 + km1.5n1.5 )(m + I)I)[20] Õ(log ɛ−1 (k0.5m3 + k-=-m1.5n1.5 )nI)[20] Õ(ɛ−2m2 Maximum concurrent flow n) as above as above Õ(ɛ−2 (k + m)m) † Õ(ɛ −2 (k + m)mn) Minimum cost concurrent flow as above Õ(ɛ −2 log ɛ −1 km 2 n 3 I)[25] Õ... |

12 |
New algorithms for generalized network flows
- Cohen, Megiddo
(Show Context)
Citation Context ...improved upon in [15, 16, 29] with the fastest algorithm developed so far by Goldfarb, Jin, and Orlin [16]. For generalized maximum flow, researchers have also developed fast approximation schemes in =-=[7, 28, 32]-=-. Very recently, Wayne [36] proposed the first polynomial combinatorial algorithms for generalized minimum cost flow. There are no known exact polynomial combinatorial algorithms for generalized multi... |

12 | Polynomial-time highest-gain augmenting path algorithms for the generalized circulation problem
- Goldfarb, Jin, et al.
- 1997
(Show Context)
Citation Context ...ructure of the problem. Goldberg, Plotkin, and Tardos [14] designed the first polynomial-time combinatorial algorithms for generalized maximum flow. Their algorithms were refined and improved upon in =-=[15, 16, 29]-=- with the fastest algorithm developed so far by Goldfarb, Jin, and Orlin [16]. For generalized maximum flow, researchers have also developed fast approximation schemes in [7, 28, 32]. Very recently, W... |

10 | Combinatorial Approximation Algorithms for Generalized Flow Problems
- Oldham
- 1999
(Show Context)
Citation Context ...berg [13] proposed a faster randomized version; Radzik [27] derandomized it. Garg and Könemann [11] simplified the method for packing problems, drawing on ideas from Young [38]. Very recently, Oldham=-= [25] pr-=-oposed FPTAS’s for a variety of generalized flow problems, using the fractional packing framework of Garg and Könemann [11]. When this framework is applied to traditional network flow, each iterati... |

9 | On max flows with gains and pure min-cost flows - Truemper - 1977 |

9 |
Faster approximation algorithms for generalized flow
- Wayne, Fleischer
- 1999
(Show Context)
Citation Context ...demands dj, 1 ≤ j ≤ k, find the maximum λ and a corresponding generalized flow that delivers λdj units of flow to tj by sending flow from sj, foreachj. ∗ An extended abstract of this paper app=-=ears in [37]. ��-=-� Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027. Email: lisa@ieor.columbia.edu. ‡ Computer Science Department, Princeton University, Princeton... |

8 | Faster algorithms for the generalized network flow problem - Radzik - 1993 |

6 | Netform modeling and applications - Glover, Klingman, et al. - 1990 |

6 |
A faster combinatorial algorithm for the generalized circulation problem
- Goldfarb, Jin
(Show Context)
Citation Context ...ructure of the problem. Goldberg, Plotkin, and Tardos [14] designed the first polynomial-time combinatorial algorithms for generalized maximum flow. Their algorithms were refined and improved upon in =-=[15, 16, 29]-=- with the fastest algorithm developed so far by Goldfarb, Jin, and Orlin [16]. For generalized maximum flow, researchers have also developed fast approximation schemes in [7, 28, 32]. Very recently, W... |

5 |
One-pass algorithms for some generalized network problems
- Charnes, Raike
- 1966
(Show Context)
Citation Context ...hat the generalized shortest path is a simple s-t path, since there can be no GAP’s. For this case, we describe a more efficient Dijkstra-like algorithm, similar to that proposed by Charnes and Raik=-=e [5]-=-, to find such a path. The difference between this approach and the approach required in the setting with flow generating cycles is analogous to the difference between the traditional shortest path pr... |

5 |
Approximate generalized circulation
- Radzik
- 1993
(Show Context)
Citation Context ...improved upon in [15, 16, 29] with the fastest algorithm developed so far by Goldfarb, Jin, and Orlin [16]. For generalized maximum flow, researchers have also developed fast approximation schemes in =-=[7, 28, 32]-=-. Very recently, Wayne [36] proposed the first polynomial combinatorial algorithms for generalized minimum cost flow. There are no known exact polynomial combinatorial algorithms for generalized multi... |

1 | Netform modeling and applications. Interfaces - Glover, Klingman, et al. - 1990 |