## Optimization with extremal dynamics

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Venue: | Physical Review Letters |

Citations: | 35 - 2 self |

### BibTeX

@ARTICLE{Boettcher_optimizationwith,

author = {Stefan Boettcher and Allon G. Percus},

title = {Optimization with extremal dynamics},

journal = {Physical Review Letters},

year = {},

pages = {2001}

}

### Years of Citing Articles

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### Abstract

We explore a new general-purpose heuristic for finding high-quality solutions to hard optimization problems. The method, called extremal optimization, is inspired by self-organized criticality, a concept introduced to describe emergent complexity in physical systems. Extremal optimization successively replaces extremely undesirable variables of a single sub-optimal solution with new, random ones. Large fluctuations ensue, that efficiently explore many local optima. With only one adjustable parameter, the heuristic’s performance has proven competitive with more elaborate methods, especially near phase transitions which are believed to coincide with the hardest instances. We use extremal optimization to elucidate the phase transition in the 3-coloring problem, and we provide independent confirmation of previously reported extrapolations for the ground-state energy of±J spin glasses in d = 3 and 4. PACS number(s): 02.60.Pn, 05.65.+b, 75.10.Nr, 64.60.Cn. Many natural systems have, without any centralized organizing facility, developed into complex structures that optimize their use of resources in sophisticated ways [1]. Biological evolution has formed efficient

### Citations

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(Show Context)
Citation Context ...ing analysis for random graphs seems to justify that expectation [13]. In an extensive numerical study on random and geometric graphs [11] we have shown that �-EO outperforms simulated annealing (SA) =-=[19,20]-=- significantly near phase transitions [21], where cutsizes first become nonzero. To this end, we have compared the averaged best results obtained for both methods for a large number of instances for i... |

460 |
A multilevel algorithm for partitioning graphs, in: Supercomputing '95
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(Show Context)
Citation Context ...e dislocated collectively to accommodate the frustrated © 2003 Wiley Periodicals, Inc. COMPLEXITY 59sTABLE 1 Best Cutsizes (and Allowed Runtime) for a Testbed of Large Graphs Large Graph GA �-EO Ref. =-=[16]-=- p-METIS Hammond (n � 4720; c � 5.8) 90 (1s) 90 (42s) 97 (8s) 92 (0s) Barth5 (n � 15606; c � 5.8) 139 (44s) 139 (64s) 146 (28s) 151 (0.5s) Brack2 (n � 62632; c � 11.7) 731 (255s) 731 (12s) — 758 (4s) ... |

306 | A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices, V4.0
- Karypsis, Kumar
- 1998
(Show Context)
Citation Context ... Comparison data for three of the large graphs are due to results from heuristics in Ref. [16] (at 50 MHz). METIS is a partitioning program based on hierarchical decomposition instead of local search =-=[17]-=-, obtaining extremely fast deterministic results (at 200 MHz). (fast) variables before the system as a whole can improve its state. In this highly correlated state, slow variables block the progressio... |

202 |
How Nature Works
- Bak
- 1996
(Show Context)
Citation Context ... annealing; punctuated equilibrium Many natural systems have, without any centralized organizing facility, developed into complex structures that optimize their use of resources in sophisticated ways =-=[1]-=-. Biological evolution has formed efficient and strongly interdependent networks in which resources *This article is a US Government work and, as such, is in the public domain in the United States of ... |

174 |
Punctuated Equilibrium and Criticality in a simple Model of Evolution, Phys
- Bak, Sneppen
- 1993
(Show Context)
Citation Context ...ous models relying on extremal processes have been proposed to explain the phenomenon of self-organization [4]. In particular, the Bak-Sneppen model of biological evolution is based on this principle =-=[5,6]-=-. Assuming an unspecified interdependency between species, it produces salient nontrivial features of paleontological data such as broadly distributed lifetimes of species, large extinction events, an... |

129 |
On the computational complexity of Ising spin glass models
- Barahona
- 1982
(Show Context)
Citation Context ...earest-neighbor pairs of spins. Due to frustration [9], ground state configurations S min are hard to find, and it has been shown that for d � 2 the problem is among the hardest optimization problems =-=[10]-=-. To find near-optimal solutions for a particular optimization problem, EO performs a neighborhood search on a single configuration S � �. As in the spin problem in Eq. (1), S consists of a large numb... |

40 |
Cavity method at zero temperature
- Mézard, Parisi
(Show Context)
Citation Context ...ndependent confirmation of those results, with far less parameter tuning. More recently [25] we have used EO to test theoretical predictions for the T � 0 properties of spin glasses on Bethe lattices =-=[26]-=-. Ref. [26] proposed a replica symmetry breaking (1RSB) solution for the ground state energy which significantly differs from replica symmetric (RS) results. As Figure 3 shows, the EO results are cons... |

33 | Memetic algorithms and the fitness landscape of the graph bi-partitioning problem, Lect
- Merz, Freisleben
- 1998
(Show Context)
Citation Context ...44s) 139 (64s) 146 (28s) 151 (0.5s) Brack2 (n � 62632; c � 11.7) 731 (255s) 731 (12s) — 758 (4s) Ocean (n � 143437; c � 5.7) 464 (1200s) 464 (200s) 499 (38s) 478 (6s) GA results are the best reported =-=[15]-=- (at 300 MHz). �-EO results are from our runs (at 200 MHz). Comparison data for three of the large graphs are due to results from heuristics in Ref. [16] (at 50 MHz). METIS is a partitioning program b... |

30 | Nature’s way of optimizing
- Boettcher, Percus
- 2000
(Show Context)
Citation Context ...ble. Although coevolution may not have optimization as its exclusive goal, it serves as a powerful paradigm. We have used it as motivation for a new approach to approximate hard optimization problems =-=[8]-=-. The heuristic we have introduced, called extremal optimization (EO), follows the spirit of the Bak-Sneppen model, updating those variables that have among the “worst” values in a solution and replac... |

27 |
Avalanche dynamics in evolution, growth, and depinning models, Phys
- Paczuski, Maslov, et al.
- 1996
(Show Context)
Citation Context ...is process prevents the inflexibility inevitable in a controlled breeding of the “good.” Various models relying on extremal processes have been proposed to explain the phenomenon of self-organization =-=[4]-=-. In particular, the Bak-Sneppen model of biological evolution is based on this principle [5,6]. Assuming an unspecified interdependency between species, it produces salient nontrivial features of pal... |

17 |
Extremal Optimization and Graph Partitioning at the Percolation Threshold
- Boettcher
- 1999
(Show Context)
Citation Context ...ve shown that this basic algorithm is very competitive for optimization problems where EO can choose randomly among many S� � N(S) that satisfy step (2c), as is the situation for graph bipartitioning =-=[8,11]-=-. But in cases such as the single spin-flip neighborhood for the spin Hamiltonian, focusing on only the worst fitness [step (2b)] leads to a deterministic process, leaving no choice in step (2c): If t... |

16 |
Extremal optimization for graph partitioning
- Boettcher, Percus
- 2001
(Show Context)
Citation Context ... rank gets excluded from further evolution, while still maintaining a bias against variables with bad fitness. In all problems studied so far, a value of � � 1 � 1/ln n �n 3 �� (6) seems to work best =-=[12,13]-=-. We have studied a simple model problem for which the asymptotic behavior of �-EO can be solved exactly [14]. The model reproduces Eq. (6) exactly in cases where the model develops a “jam” amongst it... |

13 | S.: “Valley Structures” in the phase space of a finite 3D Ising spin glass with ±i interactions
- Klotz, Kobe
- 1994
(Show Context)
Citation Context ...reezes into a local minimum, it is also well suited to enumerate the distribution of near-optimal states of a system, which is of great importance for the low-temperature dynamics in glassy materials =-=[27,28]-=-. Thus, we have also counted all lowest-energy states that EO could find for the Bethe lattice up to n � 256 (when memory became exhausted). The data extrapolate to a rather low 60 COMPLEXITY © 2003 W... |

8 | The ground state energy of the Edwards-Anderson Ising spin glass with a hybrid genetic algorithm - Pál - 1996 |

6 | Jamming model for the extremal optimization heuristic
- Boettcher, Grigni
- 2002
(Show Context)
Citation Context ...st :� S. 58 COMPLEXITY © 2003 Wiley Periodicals, Inc.sFIGURE 1 Plot of the average costs obtained by EO for a � J spin glass (left), for graph bipartitioning (center), and for the solvable model from =-=[14]-=-, all as a function of �. In the solvable model, we merely had to average runs for each n over initial conditions. For the real problems a number of instances were generated at each n. For each instan... |

5 |
Optimization by simulated annealing—an experimental evaluation; part 2, graph-coloring and number partitioning
- Johnson, Aragon, et al.
- 1991
(Show Context)
Citation Context ...ing analysis for random graphs seems to justify that expectation [13]. In an extensive numerical study on random and geometric graphs [11] we have shown that �-EO outperforms simulated annealing (SA) =-=[19,20]-=- significantly near phase transitions [21], where cutsizes first become nonzero. To this end, we have compared the averaged best results obtained for both methods for a large number of instances for i... |

4 |
private communication
- Liers
(Show Context)
Citation Context ...ve run our implementation also on two 3d lattice instances, toruspm38-50 and toruspm3-15-50, with n � 8 3 and n � 15 3 , considered in the 7th DIMACS challenge for semi-definite problems [29]. Bounds =-=[30]-=- on the ground-state cost established for the larger instance are C lower ��6138.02 (from semidefinite programming) and C upper ��5831 (from branchand-cut). EO found C(S best) ��6049 (or C/n ��1.7923)... |

3 |
Power-law distribution of landslides in an experiment on the erosion of a granular pile
- Somfai, Czirok, et al.
- 1994
(Show Context)
Citation Context ... public domain in the United States of America. rarely go to waste. Even the morphology of inanimate landscapes exhibits patterns that seem to serve a purpose, such as the efficient drainage of water =-=[2,3]-=-. Natural systems that exhibit self-organizing qualities often possess a common feature: a large number of strongly coupled entities with similar properties. Hence, at some coarse level they permit a ... |

3 |
Models of fractal river basins
- See, Cieplak, et al.
- 1998
(Show Context)
Citation Context ... public domain in the United States of America. rarely go to waste. Even the morphology of inanimate landscapes exhibits patterns that seem to serve a purpose, such as the efficient drainage of water =-=[2,3]-=-. Natural systems that exhibit self-organizing qualities often possess a common feature: a large number of strongly coupled entities with similar properties. Hence, at some coarse level they permit a ... |

3 |
Exact results for spatiotemporal correlations in a self-organized critical model of punctuated equilibrium
- Boettcher, Paczuski
- 1996
(Show Context)
Citation Context ...ous models relying on extremal processes have been proposed to explain the phenomenon of self-organization [4]. In particular, the Bak-Sneppen model of biological evolution is based on this principle =-=[5,6]-=-. Assuming an unspecified interdependency between species, it produces salient nontrivial features of paleontological data such as broadly distributed lifetimes of species, large extinction events, an... |

3 |
Searching complex state spaces with extremal optimization and other stochastic techniques
- Dall
- 2000
(Show Context)
Citation Context ... on the average rate of convergence toward better-cost configurations as a function of runtime t indicate power-law convergence, roughly like C(S best) t � C(S min) � At �0.4 [13], also found by Ref. =-=[18]-=-. Of course, it is not easy to assert for graphs of large n that those runs in fact converge closely to the optimum C(S min), but finite-size scaling analysis for random graphs seems to justify that e... |

2 | Evidence for existence of many pure ground states - Hartmann - 1997 |

2 |
Numerical results for ground states of mean-field spin glasses at low connectivities
- Boettcher
(Show Context)
Citation Context ... � 3 and e 4(�) � 2.093(1) for d � 4. Both values are consistent with the findings of Refs. [22–24], providing independent confirmation of those results, with far less parameter tuning. More recently =-=[25]-=- we have used EO to test theoretical predictions for the T � 0 properties of spin glasses on Bethe lattices [26]. Ref. [26] proposed a replica symmetry breaking (1RSB) solution for the ground state en... |

2 |
Ground-state behavior of the three-dimensional
- Hartmann
- 1999
(Show Context)
Citation Context ...reezes into a local minimum, it is also well suited to enumerate the distribution of near-optimal states of a system, which is of great importance for the low-temperature dynamics in glassy materials =-=[27,28]-=-. Thus, we have also counted all lowest-energy states that EO could find for the Bethe lattice up to n � 256 (when memory became exhausted). The data extrapolate to a rather low 60 COMPLEXITY © 2003 W... |

1 | Calculation of ground-state behavior of four-dimensional � J Ising spin glasses - Hartmann |