Survey on computational complexity with phase transitions and extremal optimization

Abstract

Applying statistical mechanics to search problems in AI, decisions and optimization has been one of the powerful channels to solve NP-hard problems. Extensive analytical and experimental research has shown that the "phase transition" phenomenon in search space is often associated with the hardness of complexity. A Bak-Sneppen (BS) model based general-purpose heuristic method, called extremal optimization (EO), proposed by Boettcher and Percus from physics society may perform very well, especially near the phase transitions in compared with other optimization methods, e.g., genetic algorithm and simulated annealing, etc. To actuate more extensive investigations on this new optimization approach particularly in control, computer and optimization communities, this survey reviews the latest research results from fundamental to practice about the connection between computational complexity and phase transitions. Then, further introduces the concepts, fundamentals, algorithms and applications of EO from its capability of self-organized criticality, backbone analysis and co-evolution moving to a far-from-equilibrium state. Finally, the concluding remarks with suggested future research are illustrated.

Extremal optimization for a dye vat scheduling problem

“…Basic idea of EO is to eliminate the unexpected component in the solution and generate a new one to replace it. EO shows good performance on many NP problems [12], such as graph partitioning, Traveling Salesman Problem (TSP), Ising spin, combinational auction, etc. combinatorial optimization problems.…”

Multistage extremal optimization for hard travelling salesman problem

“…During the past few decades, the applications of statistical physics have provided new insights into analyzing and solving combinatorial optimization problems, which has been an attractive multidisciplinary research field in physics and computer science [8][9][10][11][12][13][14]37]. The well-known algorithmic examples include classical simulated annealing (SA) [15], threshold accepting [16], Tsallis statistics [10,16,17], thermal cycling [18], iterative partial transcription [19], waiting time method [20] and survey propagation [9] and so on.…”

An Improved Real-Coded Population-Based Extremal Optimization Method for Continuous Unconstrained Optimization Problems

“…As a consequence, the basic EO algorithm and its modified versions have been successfully applied to a variety of benchmark and real-world engineering optimization problems, such as graph partitioning [9], graph coloring [10], travelling salesman problem [11,12], maximum satisfiability (MAX-SAT) problem [13,14], and steel production scheduling [15]. The more comprehensive introduction concerning EO is referred to in the surveys [16,17].…”