## Multilevel algorithms for linear ordering problems (2007)

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Citations: | 9 - 6 self |

### BibTeX

@MISC{Safro07multilevelalgorithms,

author = {Ilya Safro and Dorit Ron and Achi Brandt},

title = {Multilevel algorithms for linear ordering problems},

year = {2007}

}

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### Abstract

Linear ordering problems are combinatorial optimization problems which deal with the minimization of different functionals in which the graph vertices are mapped onto (1, 2,..., n). These problems are widely used and studied in many practical and theoretical applications. In this paper we present a variety of linear-time algorithms for these problems inspired by the Algebraic Multigrid approach which is based on weighted edge contraction. The experimental result for four such problems turned out to be better than every known result in almost all cases, while the short running time of the algorithms enables testing very large graphs.

### Citations

637 | LEDA: A platform for combinatorial and geometric computing
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(Show Context)
Citation Context ... sweeps of Gauss-Seidel relaxation) and with no use of SA. 4. RESULTS AND RELATED WORK We have implemented and tested the algorithm using standard C++, LAPACK++ [Pozo et al. 1993] and LEDA libraries [=-=Mehlhorn and Näher 1995-=-] on Linux 1.7GHz machine. The implementation is non-parallel and not fully optimized. Previous work: the minimum linear arrangement and the minimum 2-sum problems [Safro et al. 2006a; 2006b]. We have... |

301 | University of Florida sparse matrix collection: http://www.cise.ufl.edu/research/ sparse
- Davis
- 1994
(Show Context)
Citation Context ... practical high-performance and low computational cost algorithms that will outperform existing algorithms by providing better results in less running time. For that purpose we used a known benchmark =-=[17]-=- from which we took graphs of various origins and sizes including very large instances. Our multilevel algorithm exhibit linear complexity, i.e., the computational cost is proportional to |V | + |E|. ... |

247 | A review of algebraic multigrid
- Stüben
(Show Context)
Citation Context ...rk of multilevel algorithms especially designed for linear ordering problems. Our strategy is based on the Algebraic MultiGrid scheme (AMG) [Brandt et al. 1982; 1984; Brandt 1986; Briggs et al. 2000; =-=Ruge and Stüben 1987-=-; Stüben 2001a; 2001b]. While in previous works we have developed and tested special multilevel algorithms for solving the minimum linear arrangement problem [Safro et al. 2006a] and the minimum 2-sum... |

211 |
Reducing the bandwidth of sparse symmetric matrices
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- 1969
(Show Context)
Citation Context ...orithm on the five random graphs appearing in the benchmark [29]. We compare a single run of our V-cycles with the results of the exact spectral method and with those of the Cuthill-McKee permutation =-=[16]-=- which was checked also in [14]. The results are summarized in Table 2 showing a clear advantage to our multilevel approach even for those obviously unstructured random graphs. 4.3 Workbound reduction... |

127 |
Algebraic multigrid theory: The symmetric case
- Brandt
- 1986
(Show Context)
Citation Context ...re better by 34%. In this paper we present a general framework of multilevel algorithms especially designed for linear ordering problems. Our strategy is based on the Algebraic MultiGrid scheme (AMG) =-=[6, 7, 3, 10, 33, 39, 40]-=-. While in previous works we have developed and tested special multilevel algorithms for solving the minimum linear arrangement problem [35] and the minimum 2-sum problem [36], in this article we demo... |

121 | R.: Fast multiscale image segmentation
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- 2000
(Show Context)
Citation Context ...ems of equations resulting from the discretization of partial differential equations. Lately they have been applied to various other fields, yielding for example novel methods for image segmentation [=-=Sharon et al. 2000-=-] and for the linear arrangement problem [Safro et al. 2006a]. In the context of graphs it is the Laplacian matrix that represents the related set of equations. The main difference between our approac... |

117 | A survey of graph layout problems
- Díaz, Petit, et al.
(Show Context)
Citation Context ...ation of such a matrix [21]. Linear ordering problems may also be motivated as a model used in VLSI design [13] and may be used in several biological applications, graph drawing and other fields (see =-=[18, 27, 22, 37]-=-). Commonly for general graphs (or matrices) these problems are NP-hard and their decision versions are NP-complete [20]. ∗ Corresponding author : ilya.safro@weizmann.ac.il 1sSince these problems have... |

85 |
Algebraic multigrid (AMG) for sparse matrix equations. Sparsity and its Applications
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- 1984
(Show Context)
Citation Context ...re better by 34%. In this paper we present a general framework of multilevel algorithms especially designed for linear ordering problems. Our strategy is based on the Algebraic MultiGrid scheme (AMG) =-=[6, 7, 3, 10, 33, 39, 40]-=-. While in previous works we have developed and tested special multilevel algorithms for solving the minimum linear arrangement problem [35] and the minimum 2-sum problem [36], in this article we demo... |

81 |
Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations
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- 1982
(Show Context)
Citation Context ...re better by 34%. In this paper we present a general framework of multilevel algorithms especially designed for linear ordering problems. Our strategy is based on the Algebraic MultiGrid scheme (AMG) =-=[6, 7, 3, 10, 33, 39, 40]-=-. While in previous works we have developed and tested special multilevel algorithms for solving the minimum linear arrangement problem [35] and the minimum 2-sum problem [36], in this article we demo... |

64 |
A multigrid tutorial: second edition
- Briggs, Henson, et al.
(Show Context)
Citation Context |

59 | A spectral algorithm for envelope reduction of sparse matrices
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- 1995
(Show Context)
Citation Context ...ical and practical results for the bandwidth problem, e.g., [11, 30, 12, 19], to mention just a few. However, only a small number allow tests on large inputs within a reasonable execution time, e.g., =-=[2, 15, 14]-=-. Since we believe that a fair comparison of two heuristics should include final results as well as running times, and since our algorithm is able to deal with very large instances, we have thus chose... |

55 |
Optimal linear labelings and eigenvalues of graphs
- JUVAN, MOHAR
- 1992
(Show Context)
Citation Context ...zmann.ac.il 1sSince these problems have a practical significance, many heuristic algorithms were developed in order to achieve near optimal solution. Among the most successful are spectral sequencing =-=[24]-=-, optimally oriented decomposition tree [1], multilevel based [26, 23], simulated annealing [29] and others. Some of these algorithms have proven themselves superior in solution quality while others i... |

53 |
Some simplified np-complete problems
- Garey, Johnson, et al.
- 1974
(Show Context)
Citation Context ... (see [Díaz et al. 2002; Lai and Williams 1999; Horton 1997; Shahrokhi et al. 2001]). Commonly for general graphs (or matrices) these problems are NP-hard and their decision versions are NP-complete [=-=Garey et al. 1974-=-]. Since these problems have a practical significance, many heuristic algorithms were developed in order to achieve near optimal solution. Among the most successful are spectral sequencing [Juvan and ... |

51 | General highly accurate algebraic coarsening, Elect
- Brandt
(Show Context)
Citation Context ...by employing several sweeps of relaxation, first compatible then Gauss-Seidel-like. These two types of relaxation are very similar to the above initialization: The compatible relaxation, motivated in =-=[4]-=-, improves the positions of the F -nodes one by one according to the minimization criteria above (where V ′ = V ) while keeping the positions of the seeds (C-nodes) unchanged. The Gauss-Seidel-like re... |

40 | Multiscale Scientific Computation: Review 2001”. In
- Brandt
- 2002
(Show Context)
Citation Context ...esenting the original problem, but with fewer degrees of freedom. General multilevel techniques have been successfully applied to various areas of science (e.g. physics, chemistry, engineering, etc.) =-=[5, 8]-=-. AMG methods were originally developed for solving linear systems of equations resulting from the discretization of partial differential equations. Lately they have been applied to various other fiel... |

40 | An Experimental Evaluation of a Scatter Search for the Linear Ordering Problem
- Campos, Glover, et al.
- 1999
(Show Context)
Citation Context ...th of the approaches and almost equalized the results. For complete list of results see [36]. 4.2 Bandwidth There are many different theoretical and practical results for the bandwidth problem, e.g., =-=[11, 30, 12, 19]-=-, to mention just a few. However, only a small number allow tests on large inputs within a reasonable execution time, e.g., [2, 15, 14]. Since we believe that a fair comparison of two heuristics shoul... |

39 | LAPACK++: A design overview of object-oriented extensions for high performance linear algebra
- Dongarra, Pozo, et al.
- 1993
(Show Context)
Citation Context ...le relaxation); k3 = 10 (the number of sweeps of Gauss-Seidel relaxation) and with no use of SA. 4. RESULTS AND RELATED WORK We have implemented and tested the algorithm using standard C++, LAPACK++ [=-=Pozo et al. 1993-=-] and LEDA libraries [Mehlhorn and Näher 1995] on Linux 1.7GHz machine. The implementation is non-parallel and not fully optimized. Previous work: the minimum linear arrangement and the minimum 2-sum ... |

28 |
A survey of solved problems and applications on bandwidth, edge sum, and profile of graphs
- Lai, Williams
- 1999
(Show Context)
Citation Context ...blems such as : the minimum p-sum, the workbound reduction, the wavefront, the envelope size, etc. Some problems, such as finding the minimum linear arrangement [Safro et al. 2006a] or the bandwidth [=-=Lai and Williams 1999-=-], appear in many applications designed for solving problems in the large sparse matrix computation. Some other are closely related to the problem of calculating the envelope size of a symmetric matri... |

27 |
Reducing the bandwidth of a sparse matrix with tabu search
- Martí, Laguna, et al.
(Show Context)
Citation Context ...ed. 4.1 Bandwidth There are many different theoretical and practical results for the bandwidth problem, e.g., [Campos et al. 2001; Pinana et al. 2004; Caprara and González 2005; Dueck and Jeffs 1995; =-=Martí et al. 2001-=-; Lim et al. 2006], to mention just a few. However, only a small number allow tests on large inputs within a reasonable execution time, e.g., [Barnard et al. 1995; Corso and Romani 2001b; 2001a]. Sinc... |

26 | Multi-scale algorithm for the linear arrangement problem - KOREN, HAREL |

20 | J.S.: Computing an optimal orientation of a balanced decomposition tree for linear arrangement problems
- Bar-Yehuda, Even, et al.
(Show Context)
Citation Context ...actical significance, many heuristic algorithms were developed in order to achieve near optimal solution. Among the most successful are spectral sequencing [24], optimally oriented decomposition tree =-=[1]-=-, multilevel based [26, 23], simulated annealing [29] and others. Some of these algorithms have proven themselves superior in solution quality while others in execution time. One of the most popular a... |

20 |
Multi-level approaches to discrete-state and stochastic problems
- Brandt, Ron, et al.
- 1986
(Show Context)
Citation Context ...”like tool consisting of an additional permutation which stores the Best-So-Far (BSF) observed arrangement, which is being occasionally updated by a procedure called Lowest Common Configuration (LCC) =-=[9]-=-. LCC enables the systematic accumulation of sub-permutations over a sequence of different arrangements, such that each BSF sub-permutation exhibits the best (minimal) sub-order encountered so far. Th... |

19 |
Some NP-complete problems
- Garey, Graham, et al.
- 1976
(Show Context)
Citation Context ...everal biological applications, graph drawing and other fields (see [18, 27, 22, 37]). Commonly for general graphs (or matrices) these problems are NP-hard and their decision versions are NP-complete =-=[20]-=-. ∗ Corresponding author : ilya.safro@weizmann.ac.il 1sSince these problems have a practical significance, many heuristic algorithms were developed in order to achieve near optimal solution. Among the... |

17 | Graph minimum linear arrangement by multilevel weighted edge contractions - Safro, Ron, et al. |

17 | On bipartite drawings and the linear arrangement problem - SHAHROKHI, S´YKORA, et al. - 2001 |

15 | A multilevel algorithm for wavefront reduction
- Hu, Scott
(Show Context)
Citation Context ...many heuristic algorithms were developed in order to achieve near optimal solution. Among the most successful are spectral sequencing [24], optimally oriented decomposition tree [1], multilevel based =-=[26, 23]-=-, simulated annealing [29] and others. Some of these algorithms have proven themselves superior in solution quality while others in execution time. One of the most popular and exploitable methods desi... |

15 |
GRASP and path relinking for the matrix bandwidth minimization
- Pinana, Plana, et al.
(Show Context)
Citation Context ...n results and/or corresponding running time were significantly improved. 4.1 Bandwidth There are many different theoretical and practical results for the bandwidth problem, e.g., [Campos et al. 2001; =-=Pinana et al. 2004-=-; Caprara and González 2005; Dueck and Jeffs 1995; Martí et al. 2001; Lim et al. 2006], to mention just a few. However, only a small number allow tests on large inputs within a reasonable execution ti... |

14 | Experiments on the minimum linear arrangement problem - Petit |

11 |
A heuristic bandwidth reduction algorithm
- GH, Jeffs
- 1995
(Show Context)
Citation Context ...th of the approaches and almost equalized the results. For complete list of results see [36]. 4.2 Bandwidth There are many different theoretical and practical results for the bandwidth problem, e.g., =-=[11, 30, 12, 19]-=-, to mention just a few. However, only a small number allow tests on large inputs within a reasonable execution time, e.g., [2, 15, 14]. Since we believe that a fair comparison of two heuristics shoul... |

11 | An analysis of spectral envelope reduction via quadratic assignment problems
- George, Pothen
- 1997
(Show Context)
Citation Context ...n. Some other are closely related to the problem of calculating the envelope size of a symmetric matrix or, more precisely, to the amount of work needed in the Cholesky factorization of such a matrix =-=[21]-=-. Linear ordering problems may also be motivated as a model used in VLSI design [13] and may be used in several biological applications, graph drawing and other fields (see [18, 27, 22, 37]). Commonly... |

10 |
Ph.D. thesis. development of fast numerical solvers for problems in optimization and statistical mechanics
- RON
- 1990
(Show Context)
Citation Context ... heating-cooling procedure. All these accumulated updates are thus stored at the BSF which actually provides the current calculated minimum. The complete description of the LCC algorithm is given in [=-=Ron 1990-=-; Safro et al. 2006a; Ron et al. 2005]. The entire disaggregation procedure for the minimum p-sum problem is summarized below in Algorithm 4. Algorithm 4: Disaggregation(coarse level C, fine level F) ... |

9 |
Chapter 1 : Multigrid solvers and multilevel optimization strategies
- Brandt, Ron
- 2003
(Show Context)
Citation Context ...esenting the original problem, but with fewer degrees of freedom. General multilevel techniques have been successfully applied to various areas of science (e.g. physics, chemistry, engineering, etc.) =-=[5, 8]-=-. AMG methods were originally developed for solving linear systems of equations resulting from the discretization of partial differential equations. Lately they have been applied to various other fiel... |

9 |
The Optimal Linear Arrangement Problem: Algorithms and Approximation
- Horton
- 1997
(Show Context)
Citation Context ...ation of such a matrix [21]. Linear ordering problems may also be motivated as a model used in VLSI design [13] and may be used in several biological applications, graph drawing and other fields (see =-=[18, 27, 22, 37]-=-). Commonly for general graphs (or matrices) these problems are NP-hard and their decision versions are NP-complete [20]. ∗ Corresponding author : ilya.safro@weizmann.ac.il 1sSince these problems have... |

8 |
Linear placement algorithm and applications to VLSI design
- CHENG
- 1987
(Show Context)
Citation Context ...r, more precisely, to the amount of work needed in the Cholesky factorization of such a matrix [George and Pothen 1997]. Linear ordering problems may also be motivated as a model used in VLSI design [=-=Cheng 1987-=-] and may be used in several biological applications, graph drawing and other fields (see [Díaz et al. 2002; Lai and Williams 1999; Horton 1997; Shahrokhi et al. 2001]). Commonly for general graphs (o... |

6 | Heuristic spectral techniques for the reduction of bandwidth and work-bound of sparse matrices
- CORSO, 2001b
(Show Context)
Citation Context ... ”real world instances” does not achieve good enough results, while the lower bounds based on SS are very far from the best known ordering costs. Rather poor results of the exact SS were presented in =-=[14]-=- for the minimum bandwidth problem. Better results were shown there by using different approximated SS, i.e., by calculating the second eigenvector less precisely. In fact, they have tested 19 algorit... |

5 |
Models of disordered systems
- KIRKPATRICK
- 1981
(Show Context)
Citation Context ...r functionals. Also, we suggest the use of numerical calculation rather than analytic, for instance, in calculating derivatives. Finally, our postprocessing is intensified by Simulated Annealing (SA) =-=[25]-=- which is a general method to escape local minima. In the multilevel framework SA is aimed at searching only for local changes that guarantee the preservation of large-scale solution features inherite... |

5 | An Algebraic Multigrid Based Algorithm for Bisectioning General Graphs - Ron, Wishko-Stern, et al. - 2005 |

5 | Multilevel algorithm for the minimum 2-sum problem - Safro, Ron, et al. |

3 | Laying out sparse graphs with provably minimum Bandwidth
- Caprara, Salazar-Gonzalez
(Show Context)
Citation Context ...th of the approaches and almost equalized the results. For complete list of results see [36]. 4.2 Bandwidth There are many different theoretical and practical results for the bandwidth problem, e.g., =-=[11, 30, 12, 19]-=-, to mention just a few. However, only a small number allow tests on large inputs within a reasonable execution time, e.g., [2, 15, 14]. Since we believe that a fair comparison of two heuristics shoul... |

3 |
placement algorithm and applications to VLSI design
- Linear
- 1987
(Show Context)
Citation Context ... a symmetric matrix or, more precisely, to the amount of work needed in the Cholesky factorization of such a matrix [21]. Linear ordering problems may also be motivated as a model used in VLSI design =-=[13]-=- and may be used in several biological applications, graph drawing and other fields (see [18, 27, 22, 37]). Commonly for general graphs (or matrices) these problems are NP-hard and their decision vers... |

3 |
F.: Heuristics for matrix bandwidth reduction
- Lim, Rodrigues, et al.
- 2006
(Show Context)
Citation Context ...here are many different theoretical and practical results for the bandwidth problem, e.g., [Campos et al. 2001; Pinana et al. 2004; Caprara and González 2005; Dueck and Jeffs 1995; Martí et al. 2001; =-=Lim et al. 2006-=-], to mention just a few. However, only a small number allow tests on large inputs within a reasonable execution time, e.g., [Barnard et al. 1995; Corso and Romani 2001b; 2001a]. Since we believe that... |