Abstract
This chapter analyzes the search behavior of the ABSS in detail to answer three fundamental questions for the ABSS: (1) how can we construct the edge configuration of the attractor using a small set of local search trajectories? (2) what is the relationship between the size of the constructed attractor and the size of the TSP instance? and (3) is the best tour in the attractor the optimal tour in the solution space?
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References
Aarts, E., & Lenstra, J. K. (2003). Local search in combinatorial optimization. Princeton University Press.
Abarbaanel, H. (1996). Analysis of observed chaotic data. Springer.
Abarbanel, H., Brow, R., Sidorowich, J. J., & Tsimring, L. S. (1993). The analysis of observed chaotic data in physical systems. Reviews of Modern Physics, 65(2), 1331–1392. https://doi.org/10.1103/RevModPhys.65.1331
Alligood, K. T., Sauer, T. D., & Yorke, J. A. (1997). Chaos: Introduction to dynamical systems. Springer.
Applegate, D. L., Bixby, R. E., Chvátal, V., & Cook, W. J. (2006). The traveling salesman problem: A computational study. Princeton University Press.
Arbieto, A., & Obata, D. (2015). On attractor and their basins. Involve, 8(2), 195–209. https://doi.org/10.2140/INVOLVE.2015.8.195
Arrowsmith, D. K., & Place, C. M. (1990). An introduction to dynamical systems. Cambridge University Press.
Auslander, J., Bhatia, N. P., & Seibert, P. (1964). Attractors in dynamical systems. NASA Technical Report NASA-CR-59858.
Baudet, G. M. (1978). On the branching factor of the alpha-beta pruning algorithm. Artificial Intelligence, 10(2), 173–199. https://doi.org/10.1016/0004-3702(78)80011-3
Boese, K. D., Kahng, A. B., & Muddu, S. (1994). A new adaptive multi-start technique for combinatorial global optimization. Operation Research Letters, 16(2), 101–113. https://doi.org/10.1016/0167-6377(94)90065-5
Brin, M., & Stuck, G. (2016). Introduction to dynamical systems. Cambridge University Press.
Brock, W. (1986). Distinguishing random and deterministic systems: Abridged version. Journal of Economic Theory, 40(1), 168–195. https://doi.org/10.1016/0022-0531(86)90014-1
Broomhead, D. S., & King, G. P. (1987). Extracting qualitative dynamics from experimental data. Physica D: Nonlinear Phenomena, 20(2/3), 217–236. https://doi.org/10.1016/0167-2789(86)90031-X
Brown, R. J. (2018). A modern introduction to dynamical systems. Oxford University Press.
Buescu, J. (1991). Exotic attractors. Birkhäuser.
Chandra, B., Karloff, H., & Tovey, C. (1999). New results on the old k-opt algorithm for the traveling salesman problem. SIAM Journal on Computing, 28(6), 1998–2029. https://doi.org/10.1137/S0097539793251244
Collet, P., & Eckmann, J. P. (1980). Iterated maps on the interval as dynamical systems. Birkhäuser.
Cook, W., & Seymour, P. D. (2003). Tour merging via branch decomposition. INFORMS Journal on Computing, 15(3), 233–248. https://doi.org/10.1287/ijoc.15.3.233.16078
Cover, T., & Thomas, J. (1991). Elements of information theory. Wiley.
Dantzig, G. B. (1963). Linear programming and extensions. Princeton University Press.
Dantzig, G. B., Fulkerson, R., & Johnson, S. (1954). Solution of a large-scale traveling-salesman problem. Journal of the Operations Research Society of America, 2(4), 393–410. https://doi.org/10.1287/opre.2.4.393
Dénes, A., & Makey, G. (2011). Attractors and basins of dynamical systems. Electronic Journal of Qualitative Theory of Differential Equations, 20(20), 1–11. https://doi.org/10.14232/ejgtcle.2011
Dorigo, M. (1992). Optimization, learning and natural algorithms. Ph.D thesis, Dip Electronica e Informazione, Politecnico Di Milano, Italy.
Dorigo, M., & Gambardella, L. M. (1997). Ant colony system: A cooperative learning approach to the traveling salesman problem. IEEE Transactions on Evolutionary Computation, 1(1), 53–66. https://doi.org/10.1109/4235.585892
Dorigo, M., & Stützle, T. (2004). Ant colony optimization. The MIT Press.
Edelkamp, S., & Korf, R. E. (1998). The branching factor of regular search spaces. In Proceeding of the 15th National Conference on Artificial Intelligence (pp. 292–304).
Feo, T. A., & Resende, M. G. (1995). Greedy randomized adaptive search procedure. Journal of Global Optimization, 6, 109–133. https://doi.org/10.1007/BF01096763
Fischer, S. T. (1995). A note on the complexity of local search problems. Information Processing Letters, 53(2), 69–75. https://doi.org/10.1016/0020-0190(94)00184-Z
Fogelberg, H. (1992). The phase space approach to complexity. Journal of Statistical Physics, 69, 411–425. https://doi.org/10.1007/BF01053799
Frazer, A. M., & Swinney, H. L. (1986). Independent coordinates for strange attractors from mutual information. Physical Review A, 33(2), 1134–1140. https://doi.org/10.1103/PhysRevA.33.1134
Freisleben, B., & Merz, P. (1996). A genetic local search algorithm for solving symmetric and asymmetric traveling salesman problems. In Proceedings of IEEE International Conference on Evolutionary Computation (pp. 616–621). IEEE.
Gagniuc, P. A. (2017). Markov chains: From theory to implementation and experimentation. John Wiley & Sons.
Gallager, R. G. (1968). Information theory and reliable communication. Wiley.
Glover, F. (2000). Multi-start and strategic oscillation methods—Principles to exploit adaptive memory. In M. Laguna & J. L. Gonzales-Velards (Eds.), Computing tools for modeling optimization and simulation (pp. 1–25). Kluwer.
Glover, F., & Punnen, A. P. (1997). The traveling salesman problem: New solvable cases and linkages with the development of approximation algorithms. Journal of the Operational Research Society, 48(5), 502–510. https://doi.org/10.2307/3010508
Grassberger, P., & Procaccia, I. (1983). Characterization of strange attractor. Physical Review Letters, 50(5), 346–349. https://doi.org/10.1103/PhysRevLett.50.346
Grover, L. K. (1992). Local search and the local structure of NP-complete problems. Operations Research Letters, 12(4), 235–243. https://doi.org/10.1016/0167-6377(92)90049-9
Hains, D., Whitley, D., & Howe, A. (2011). Revisiting the big valley search space structure in the TSP. Journal of Operational Research Society, 62(2), 305–312. https://doi.org/10.1057/jors.2010.116
Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biametnika, 57(1), 97–109. https://doi.org/10.1093/biomet/57.1.97
Hawkins, J. (2021). Attractors in dynamical systems. In Ergodic dynamics. Graduate texts in mathematics (Vol. 289). Springer. https://doi.org/10.1007/978-3-030-59242-4_3
Herley, M. (1982). Attractors: Persistence and density of their basins. Transactions of the American Mathematical Society, 269(1), 247–271. https://doi.org/10.1090/s0002-9947-1982-0637037-7
Hilborn, R. C. (1994). Chaos and nonlinear dynamics: An introduction for science and engineers. Oxford University Press.
Holzfuss, J., & Mayer-Kress, G. (1986). An approach to error-estimation in the application of dimension algorithm. In G. Mayer-Kress (Ed.), Dimensions and entropies in chaotic systems: Quantification of complex behavior (pp. 114–122). Springer.
Johnson, D. S. (1990). Local optimization and the traveling salesman problem. In Proceedings of the 17th International Colloquium on Automata, Languages and Programming, ICALP1990 (pp. 446–461). Springer. https://doi.org/10.1007/BFb0032050
Johnson, D. S., & McGeocj, L. A. (1995). The traveling salesman problem: A case study in local optimization. In E. Aarts & J. K. Lenstra (Eds.), Local search in combinatorial optimization. John Wiley & Sons.
Jones, T., & Forrest, S. (1995). Fitness distance correlation as a measure of problem difficulty for generic algorithm. In Proceedings of the Sixth International Conference on Genetic Algorithms (pp. 184–192). Morgan Kaufmann.
Kauffman, S., & Levin, S. (1987). Towards a general theory of adaptive walks on rugged landscapes. Journal of Theoretical Biology, 128(1), 11–45. https://doi.org/10.1016/S0022-5193(87)80029-2
Katok, A. B., & Hasselblatt, B. (1999). Introduction to the modern theory of dynamical systems. Cambridge University Press.
Kennel, M. B., Brown, R., & Abarbanel, H. D. I. (1992). Determining embedding dimension for phase-space reconstruction using a geometrical construction. Physical Review A, 45(6), 3403–3411. https://doi.org/10.1103/PhysRevA.45.3403
Kirkpatrick, S., & Toulouse, G. (1985). Configuration space analysis of traveling salesman problem. Journal de Physique, 46(8), 1277–1292. https://doi.org/10.1051/jphys:019850046080127700
Kolokoltsov, V. N. (2010). Nonlinear Markov process and kinetic equations. Cambridge University Press.
Korf, R. E. (1985). Depth-first iterative deepening: An optimal admissible tree search. Artificial Intelligence, 27(1), 97–109. https://doi.org/10.1016/0004-3702(85)90084-0
Kroese, D. P., Taimre, T., & Botev, Z. I. (2011). Handbook of Monte Carlo methods. John Wiley & Sons.
Laguna, M., & Martí, R. (1999). GRASP and path relinking for a 2-player straight line crossing minimization. INFORMS Journal on Computing, 11(1), 44–52. https://doi.org/10.1287/ijoc.11.1.44
Lin, S. (1965). Computer solution of the traveling salesman problem. The Bell System Technical Journal, 44(10), 2245–2269. https://doi.org/10.1002/j.1538-7305.1965.tb04146.x
Lin, S., & Kernighan, B. W. (1973). An effective heuristic algorithm for the traveling salesman problem. Operations Research, 21(2), 498–516. https://doi.org/10.1278/opre.21.2.498
Macken, C. A., Hagan, P. S., & Perelson, A. S. (1991). Evolutionary walks on rugged landscapes. SIAM Journal of Applied Mathematics, 51(3), 799–827. https://doi.org/10.1137/0151040
Martin, O., Otto, S. W., & Felten, E. W. (1992). Large step Markov chains for the TSP incorporating local search heuristics. Operations Research Letters, 11(4), 219–224. https://doi.org/10.1016/0167-6377(92)90028-2
Merz, P., & Freisleben, B. (1998). Memetic algorithms and the fitness landscape of the graph bi-partitioning problem. In A. E. Eiben, T. Bäck, M. Schoemauer, & H. P. Schwefel (Eds.), Parallel problem solving from nature—PPSN V, LNCS 1948 (pp. 765–774). Springer.
Merz, P., & Freisleben, B. (1999). Fitness landscapes and memetic algorithm design. In D. Corn, M. Dorigo, & F. Glover (Eds.), New ideas in optimization (pp. 245–260). McGraw-Hill.
Meyn, S., & Tweedie, R. L. (2009). Markov chains and stochastic stability. Cambridge University Press.
Mézard, M., & Parisi, G. (1986). A replica analysis of the traveling salesman problem. Journal de Physique, 47(8), 1285–1296. https://doi.org/10.1051/jphys:019860047080128500
Milnor, J. (1985). On the concept of attractor. Communications in Mathematical Physics, 99(2), 177–195. https://doi.org/10.1007/BF01212280
Milnor, J. (2010). Collected papers of John Milnor VI: Dynamical systems (1953–2000). American Mathematical Society.
Ott, E., Sauer, T., & Yorke, J. A. (1994). Co** with chaos: Analysis of chaotic data and the exploitation of chaotic systems. John Wiley & Sons.
Packard, N., Crutchfield, J. P., Farmer, J. D., & Shaw, R. S. (1980). Geometry from a time series. Physical Review Letters, 45(9), 712–716. https://doi.org/10.1103/PhysRevLett.45.712
Padberg, M., & Rinaldi, G. (1991). A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Review, 33(1), 60–100. https://doi.org/10.1137/1033004
Papadimitriou, C. H., & Steiglitz, K. (1998). Combinatorial optimization: Algorithms and complexity. Dover Publications.
Parker, T. S., & Chua, L. O. (1990). Practical numerical algorithms for chaotic systems. Springer.
Pearl, J. (1982). The solution for the branching factor of the alpha-beta pruning algorithm and its optimality. Communication of the ACM, 25(8), 559–564. https://doi.org/10.1145/358589.358616
Ramsey, J. B., & Yuan, H. J. (1987). The statistical properties of dimension calculations using small data sets. Nonlinearity, 3(1), 155–176. https://doi.org/10.1088/0951-7715/3/1/009
Reeves, C. R. (1999). Landscapes, operators and heuristic search. Annals of Operations Research, 86, 473–490. https://doi.org/10.1023/A:1018983524911
Reidys, C. M., & Stadler, P. F. (2002). Combinatorial landscapes. SIAM Review, 44(1), 3–54. https://doi.org/10.1137/S0036144501395952
Resende, M. G., Martí, R., Gallego, M., & Duarte, A. (2010). GRASP and path relinking for the max–min diversity problem. Computer & Operations Research, 37(3), 498–508. https://doi.org/10.1016/j.cor.2008.05.011
Ruelle, D. (1981). Small random perturbations of dynamical systems and the definition of attractor. Communications in Mathematical Physics, 82, 137–151. https://doi.org/10.1007/BF01206949
Sauer, T., Yorke, J. A., & Casdagli, M. (1991). Embedology. Journal of Statistical Physics, 65(3/4), 579–616. https://doi.org/10.1007/BF01053745
Shammon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379–423 & 623–656. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
Sorkin, G. B. (1991). Efficient simulated annealing on fractal energy landscapes. Algorithmica, 6, 367–418. https://doi.org/10.1007/BF01759051
Sourlas, N. (1986). Statistical mechanics and the traveling salesman problem. Europhysics Letter, 2(12), 919–923. https://doi.org/10.1209/0295-5075/2/12/006
Takens, F. (1981). Detecting strange attractors in turbulence. Lecture Notes in MathematicsIn D. Rand & L. S. Young (Eds.), Dynamical systems and turbulence, Warwick 1980 (Vol. 898, pp. 366–382). Springer.
Takens, F. (1985). On the numerical determination of the dimension of an attractor. In N. Breaksma, H. Broer, & F. Takens (Eds.), Dynamical systems and bifurcations (pp. 99–106). Springer.
Tong, H., & Lim, K. S. (1980). Threshold, autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society, 42(3), 245–292. https://doi.org/10.1111/j.2517-6161.1980.tb01126.x
Weinberger, E. D. (1990). Correlated and uncorrelated fitness landscapes and how to tell the difference. Biological Cybernetics, 63, 325–336. https://doi.org/10.1007/BF00202749
Weinberger, E. D. (1991). Local properties of Kauffman’s N-k model: A tunably rugged energy landscape. Physics Review A, 44, 6399–6413. https://doi.org/10.1103/PhysRevA.44.6399
Zhang, W. (2004). Configuration landscape analysis and backbone guided local search. Part I: Satisfiability and maximum satisfiability. Artificial Intelligence, 158(1), 1–26. https://doi.org/10.1016/j.artint.2004.04.001
Zhang, W., & Looks, M. (2005). A novel local search algorithm for the traveling salesman problem that exploits backbones. In Proceedings of 19th International Joint Conference on Artificial Intelligence (pp. 343–348). ACM.
Zweig, G. (1995). An effective tour construction and improvement procedure for the traveling salesman problem. Operations Research, 43(6), 1049–1057. https://doi.org/10.1287/opre.43.6.1049
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Li, W. (2024). Analysis of the Attractor-Based Search System. In: The Traveling Salesman Problem. Synthesis Lectures on Operations Research and Applications. Springer, Cham. https://doi.org/10.1007/978-3-031-35719-0_4
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