## Fast Approximation Algorithms for Fractional Packing and Covering Problems (1995)

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Citations: | 233 - 13 self |

### BibTeX

@MISC{Plotkin95fastapproximation,

author = {Serge A. Plotkin and David B. Shmoys and Éva Tardos},

title = {Fast Approximation Algorithms for Fractional Packing and Covering Problems},

year = {1995}

}

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### Abstract

This paper presents fast algorithms that find approximate solutions for a general class of problems, which we call fractional packing and covering problems. The only previously known algorithms for solving these problems are based on general linear programming techniques. The techniques developed in this paper greatly outperform the general methods in many applications, and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems. Our algorithm is a Lagrangean relaxation technique; an important aspect of our results is that we obtain a theoretical analysis of the running time of a Lagrangean relaxation-based algorithm. We give several applications of our algorithms. The new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the non-preemptive models, for the job shop problem, for th...

### Citations

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279 |
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Citation Context ...solutions and thus yield theorems relating the linear and fractional optima along the lines of Raghavan & Thompson [24] and give alternative deterministic algorithms to obtain the results of Raghavan =-=[23]-=-. The modified algorithm is, in some cases, more efficient than the original algorithm, due to the fact that it terminates as soon as it can no longer improve the current solution while maintaining in... |

239 |
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Citation Context ...ge ij 2 EH . The dilation of the embedding is the maximum number of edges on one of the paths used, and the congestion is the maximum number of paths that contain the same edge in G. Leighton and Rao =-=[20]-=- gave an algorithm to embed H in G with dilation and congestion both O( log N ff ). If H is an expander, and hence each subset S of at most N=2 nodes has \Omega\Gamma jSj) edges leaving it in H, then ... |

203 | Approximation algorithms for scheduling unrelated parallel machines
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Citation Context ...ften denoted RjjC max , is NP-complete, and in fact, Lenstra, Shmoys, & Tardos showed that there does not exist an ffl-approximation algorithm with ffl ! 1=2 unless P = NP . Lenstra, Shmoys, & Tardos =-=[21]-=- also gave a 1-approximation algorithm for it, based on a 1-relaxed decision procedure. If there exists a schedule of length T , then the following linear program has a feasible solution: N X j=1 p ij... |

170 | Improved Approximation Algorithms for the Multi-Commodity Flow Problem and Local Competitive Routing
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- 1994
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Citation Context ...ind approximate solutions to multicommodity flow problems, first by Shahrokhi & Matula [25], and later by Klein, Plotkin, Stein & Tardos [16] and Leighton, Makedon, Plotkin, Stein, Tardos & Tragoudas =-=[19]-=-. Recently, extensions of this method to other applications were found independently by Grigoriadis & Khachiyan [10]. An important theoretical aspect of our results is their connection to Lagrangean r... |

150 |
The maximum concurrent flow problem
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- 1990
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Citation Context ...t f(n) log c ns\Omega\Gamma g(n)); we define O analogously. Our approach extends a method previously applied to find approximate solutions to multicommodity flow problems, first by Shahrokhi & Matula =-=[25]-=-, and later by Klein, Plotkin, Stein & Tardos [16] and Leighton, Makedon, Plotkin, Stein, Tardos & Tragoudas [19]. Recently, extensions of this method to other applications were found independently by... |

121 |
An efficient approximation scheme for the one-dimensional bin-packing problem
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Citation Context ... it as well as its linear relaxation [5, 6]. This linear relaxation is also the key ingredient of the fully polynomial approximation scheme for the bin-packing problem that is due to Karmarkar & Karp =-=[13]-=-. We also consider problems with simultaneous packing and covering constraints. In this paper we focus on obtaining approximate solutions for these problems. For an error parameter ffl ? 0, a point x ... |

120 |
A new algorithm for minimizing convex functions over convex sets
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- 1996
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Citation Context ...r every vertex of P , since the separation subroutine for the dual problem can be solved with the optimization subroutine for P . The problem can be solved more efficiently by the algorithm of Vaidya =-=[27]-=-; it obtains the optimal value in O(mL) calls to an optimization subroutine for P plus O(mM(m)L) additional time; L and M(m) denote the binary size of the problem and the time needed to invert m by m ... |

118 |
The traveling-salesman problem and minimum spanning trees
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(Show Context)
Citation Context ...ws of commodity `, and Axsb describes the joint capacity constraints. In Section 5, we shall discuss the following further applications: the Held & Karp lower bound for the traveling salesman problem =-=[11]-=- (packing 1-trees subject to degree-2 and cost constraints); scheduling unrelated parallel machines in both the preemptive and non-preemptive models, as well as scheduling job shops (packing jobs subj... |

101 |
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Citation Context ...o cover that demand using as few raws as possible. Gilmore & Gomory proposed a natural integer programming formulation of this problem and studied methods to solve it as well as its linear relaxation =-=[5, 6]-=-. This linear relaxation is also the key ingredient of the fully polynomial approximation scheme for the bin-packing problem that is due to Karmarkar & Karp [13]. We also consider problems with simult... |

87 |
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(Show Context)
Citation Context ...can be used, and we wish to fill our knapsack as profitably as possible. Although this is NP -hard, recall that by Theorem 3.10 an ffl=2-approximation algorithm would suffice for our purposes. Lawler =-=[17]-=- gave efficient approximation algorithms for the M-piece ordinary knapsack problem that run in O(M ffl \Gamma2 ) and O(M log ffl \Gamma1 + ffl \Gamma4 ) time. Next we adapt both algorithms for the abo... |

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 ...logously. Our approach extends a method previously applied to find approximate solutions to multicommodity flow problems, first by Shahrokhi & Matula [25], and later by Klein, Plotkin, Stein & Tardos =-=[16]-=- and Leighton, Makedon, Plotkin, Stein, Tardos & Tragoudas [19]. Recently, extensions of this method to other applications were found independently by Grigoriadis & Khachiyan [10]. An important theore... |

82 | Improved approximation algorithms for shop scheduling problems
- Shmoys, Stein, et al.
- 1994
(Show Context)
Citation Context ...ax j P i p ij , the maximum total processing time of any job, and finally, let \Pi max denote max i P j;k:m kj =i p kj , the maximum total processing time assigned to a machine. Shmoys, Stein, & Wein =-=[26] giv-=-e a randomized O(log 2 (M + ��))-approximation algorithm for this problem and a deterministic variant that uses the randomized rounding technique of Raghavan & Thompson [24] and its deterministic ... |

59 |
Fast approximation schemes for convex programs with many block and coupling constraints
- Grigoriadis, Khachiyan
- 1994
(Show Context)
Citation Context ...tkin, Stein & Tardos [16] and Leighton, Makedon, Plotkin, Stein, Tardos & Tragoudas [19]. Recently, extensions of this method to other applications were found independently by Grigoriadis & Khachiyan =-=[10]-=-. An important theoretical aspect of our results is their connection to Lagrangean relaxation. The main idea of our algorithm is as follows. We maintain a point x 2 P that does not satisfy Axsb, and r... |

48 |
Probabilistic recurrence relations
- Karp
- 1994
(Show Context)
Citation Context ...e k X s=1 \Delta ssmin ae ffl 8 P ` ae ` ; ff 2k oe ffl\Phi: Since ffs2 \Gamma1 ffl \Gamma1 ln(2mffl \Gamma1 ), the claimed bound on the expected decrease of \Phi follows. We use a result due to Karp =-=[15]-=- to analyze the number of iterations used by the randomized version of Improve-Packing. Let ffi \Phi denote the ratio of upper and lower bounds on the potential function \Phi during a single execution... |

48 | A faster algorithms for finding the minimum cut in a graph, in - Hao, Orlin - 1992 |

46 | An O(n ) algorithm for minimum cuts - Karger, Stein - 1993 |

40 | On preemptive scheduling of unrelated parallel processors by linear programming
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(Show Context)
Citation Context ...mega (MN) factors, respectively. In a related model, we consider schedules with preemptions: a job may be started on one machine, interrupted, and then continued later on another. Lawler & Labetoulle =-=[18]-=- showed that an optimal preemptive schedule for this problem, RjpmtnjC max , can be found by minimizing T subject to N X j=1 p ij x ijsT; i = 1; : : : ; M; (19) M X i=1 p ij x ijsT; j = 1; : : : ; N; ... |

34 |
Speeding up linear programming using fast matrix multiplication
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(Show Context)
Citation Context ...n algorithm is obtained by using Vaidya's algorithm [27] to approximately solve the linear programming dual of the problem, and then use the techniques of Karmarkar & Karp and the algorithm of Vaidya =-=[28]-=- to obtain a primal solution. The resulting deterministic algorithm runs in O (M 4 M(M)+ M 3 ffl \Gamma2 ) time. A randomized version runs in O (M 3 M(M) + M 3 ffl \Gamma2 ) time. As for our algorithm... |

29 |
Fast algorithms for convex quadratic programming and multicommodity flows
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Citation Context ...ng algorithm of Vaidya [28] for problems where the polytope P can be described with few variables. Furthermore, if the problem has an appropriate network structure then the ideas of Kapoor and Vaidya =-=[12]-=- can be used to speed up the matrix inversions involved. The algorithm in [27] obtains an optimal dual solution to a fractional packing or covering problem, but no primal solution. By a perturbation a... |

18 |
An O(n) algorithm for the multiple-choice knapsack linear program
- Dyer
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(Show Context)
Citation Context ...t ae js1. To optimize over P j , note that this is the dual of a 2-variable linear program with M constraints, and, in fact, it is a fractional multiple-choice knapsack problem with M variables. Dyer =-=[1]-=- has shown that this problem can be solved in O(M) time. For the deterministic version, when P = P 1 \Theta \Delta \Delta \Delta \Theta P N , we have aesN . To optimize over P , we solve N disjoint mu... |

15 |
A natural randomization strategy for multicommodity flow and related algorithms
- Goldberg
- 1992
(Show Context)
Citation Context ...ssary to compute CP (y). This is no longer done each iteration; instead, this condition is checked with probability 1=k. This particular method of randomizing is an extension of an idea that Goldberg =-=[7]-=- has used for the multicommodity flow problem, and was also independently discovered by Grigoriadis & Khachiyan [10]. The key to the randomized version of our algorithm is the following lemma. The pro... |

12 | Using interior-point methods for fast parallel algorithms for bipartite matching and related problems
- Goldberg, Plotkin, et al.
- 1992
(Show Context)
Citation Context ... some applications, such as the bipartite matching problem, it is possible to find a starting solution with L = log(nffl \Gamma1 ), which is, roughly speaking, the number of bits of accuracy required =-=[9]-=-. Unfortunately, we do not know of comparable results for any of our applications. For two of our applications, the bin-packing problem and the Held-Karp bound, such results would yield algorithms wit... |

7 |
The trim problem
- Eisemann
- 1957
(Show Context)
Citation Context ...want an integer solution, the linear relaxation of this formulation has been extremely useful in practical applications; furthermore, there are applications in which patterns may be used fractionally =-=[2]-=-. Also, finding an approximate solution to this linear relaxation is the key ingredient of Karmarkar & Karp's [13] fully polynomial approximation scheme for the bin-packing problem. Given a possible n... |

3 |
Using Euler partitions to edge-color bipartite multi-graphs
- Gabow
- 1976
(Show Context)
Citation Context ...d these rescaled times by forming the multigraph �� G, where each ij 2 E occurs with multiplicity d��p ij e. Thus, the maximum degree \Delta of this graph is at most Q+N . Using an algorithm o=-=f Gabow [4]-=-, we can color this graph in O(M \Delta log \Delta) time with 2 dlog 2 \Deltae colors. By choosing Q = 2 l \Gamma N , where l = dlog 2 (N + N ffl \Gamma1 )e, it follows that \Deltas2 l = O(N=ffl). Eac... |

2 |
de Velde. Machine scheduling and Lagrangian relaxation. Doctoral thesis
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(Show Context)
Citation Context ...mputing the minimum modified processing time y i p ij , where the minimization is restricted to those machines for which p ijsT . This approach is quite similar to the ascent method that Van de Velde =-=[29]-=- used to solve this linear program; he also used a Lagrangean method that, in each iteration, constructs a schedule by assigning each job to its fastest machine with respect to the modified processing... |

1 |
Combinatorial algorithms for the generalized flow problem
- Goldberg, Plotkin
- 1990
(Show Context)
Citation Context ...omized analog than runs in O(MN log M log N) expected time. The fastest previously known algorithm for solving this problem is the Fat-Path generalized flow algorithm of Goldberg, Plotkin, and Tardos =-=[8]-=-. In order to convert the packing problem defined by (15-18) into a generalized flow problem, we construct a bipartite graph with nodes representing jobs and machines and introduce an edge from machin... |

1 | An ~ O(n ) algorithm for minimum cut. Unpublished manuscript - Karger, Stein - 1992 |