Abstract
Four known search strategies used in branch-and-bound algorithms-heuristic search, depth-first search, best-bound search, and breadth-first search-are theoretically compared from the viewpoint of the performance of the resulting algorithms. Heuristic search includes the other three as special cases. Since heuristic search is determined by a heuristic functionh, we first investigate how the performance of the resulting algorithms depends onh. In particular, we show that heuristic search is stable in the sense that a slight change inh causes only a slight change in its performance. The “best” and the “worst” heurstic functions are clarified, and also discussed is how the heuristic functionh should be modified to obtain a branch-and-bound algorithm with an improved performance. Finally, properties and limitations of depth-first search, best-bound search, and breadth-first search viewed as special cases of heuristic search are considered. In particular, it is shown that the stability observed for heuristic search no longer holds for depth-first search.
Similar content being viewed by others
References
N. Agin, “Optimum Seeking with Branch and Bound,”Manage. Sci. 13:B176-B185 (1966).
E. Balas, “A Note on the Branch-and-Bound Principle,”Oper. Res. 16:442–445 (1968).
E. Balas, “Discrete Programming by the Filter Method,”Oper. Res. 15:915–957 (1967).
M. Benichou, J. M. Gauthier, P. Girodet, G. Hentges, G. Ribiere, and O. Vincent, “Experiments in Mixed-Integer Linear Programming,”Math. Program. 1:76–94 (1971).
J. J. H. Forrest, J. P. H. Hirst, and J. A. Tomlin, “Practical Solution of Large Mixed Integer Programming Problems with umpire,”Manage. Sci. 20:736–773 (1974).
B. L. Fox and L. E. Schrage, “The Values of Various Strategies in Branch-and-Bound,” Technical Report, Graduate School of Business, University of Chicago (1972).
A. M. Geoffrion, “Integer Programming by Implicit Enumeration and Balas Method,”SIAM Rev. 9:178–190 (1967).
F. Glover, “A Multiphase-Dual Algorithm for the Zero-One Integer Programming Problem,”Oper. Res. 13:879–919 (1965).
S. W. Golomb and L. D. Baumert, “Backtrack Programming,”JACM 12:516–524 (1965).
P. E. Hart, N. J. Nilsson, and B. Raphael, “A Formal Basis for the Heuristic Determination of Minimal Cost Paths,”IEEE Trans. Sys. Sci. Cybern. SSC-4:100–107 (1968).
T. Ibaraki, “A Generalization of Depth-First Search in Branch-and-Bound Algorithms,” Working Paper, Department of Applied Mathematics and Physics, Kyoto University (1974).
T. Ibaraki, “The Power of Dominance Relations in Branch-and-Bound Algorithms,” Working Paper, Department of Applied Mathematics and Physics, Kyoto University,JACM, to appear (1975).
T. Ibaraki, “On the Computational Efficiency of Branch-and-Bound Algorithms,” Working Paper, Department of Applied Mathematics and Physics, Kyoto University (1975).
W. H. Kohler and K. Steiglitz, “Characterization and Theoretical Comparison of Branch-and-Bound Algorithms for Permutation Problems,”JACM 21:140–156 (1974).
E. L. Lawler and D. E. Wood, “Branch-and-Bound Method: A Survey,”Oper. Res. 14:699–719 (1966).
L. G. Mitten, “Branch-and-Bound Method: General Formulation and Properties,”Oper. Res. 18:24–34 (1970).
N. J. Nilsson,Problem-Solving Methods in Artificial Intelligence (McGraw-Hill, New York, 1971).
I. Pohl, “First Results on the Effect of Error in Heuristic Search,” inMachine Intelligence 5, B. Meltzer and D. Michie, Eds. (Edinburgh University Press, 1969).
S. K. Sahni,“Algorithms for Scheduling Independent Tasks,”JACM 23:114–127(1976).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ibaraki, T. Theoretical comparisons of search strategies in branch-and-bound algorithms. International Journal of Computer and Information Sciences 5, 315–344 (1976). https://doi.org/10.1007/BF00998631
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00998631