Analyzing the Complexity of Finding Good Neighborhood Functions for Local Search Algorithms
A drawback to using local search algorithms to address NP-hard discrete optimization problems is that many neighborhood functions have an exponential number of local optima that are not global optima (termed L-locals). A neighborhood function η is said to be stable if the number of L-locals is polynomial. A stable neighborhood function ensures that the number of L-locals does not grow too large as the instance size increases and results in improved performance for many local search algorithms. This paper studies the complexity of stable neighborhood functions for NP-hard discrete optimization problems by introducing neighborhood transformations. Neighborhood transformations between discrete optimization problems consist of a transformation of problem instances and a corresponding transformation of solutions that preserves the ordering imposed by the objective function values. In this paper, MAX Weighted Boolean SAT (MWBS), MAX Clause Weighted SAT (MCWS), and Zero-One Integer Programming (ZOIP) are shown to be NPO-complete with respect to neighborhood transformations. Therefore, if MWBS, MCWS, or ZOIP has a stable neighborhood function, then every problem in NPO has a stable neighborhood function. These results demonstrate the difficulty of finding effective neighborhood functions for NP-hard discrete optimization problems.
Subject Classificationanalysis of algorithms computational complexity
KeywordsComputational complexity Local search algorithms NP-hard discrete optimization problems
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- Aarts E., Lenstra J.K. ed. (1997), Local Search in Combinatorial Optimization. John Wiley & Sons, ChichesterGoogle Scholar
- Armstrong D.E. (2002), A local search algorithm approach to analyzing the complexity of discrete optimization problems, Ph.D. Dissertation, University of Illinois, Urbana, ILGoogle Scholar
- Ausiello, G., Crescenzi P. and Protasi, M. (1995), Approximate solution of NP optimization problems, Theoretical Computer Science 150, 1–55.Google Scholar
- Ausiello G., Crescenzi P., Gambosi G., Kann, Marchetti-Spaccamela A., Protasi M. (1999), Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer-Verlag, BerlinGoogle Scholar
- Cook, S.A. (1971), The complexity of theorem-proving procedures, Proceedings of 3rd ACM Symposium on Theory of Computation, 151–158.Google Scholar
- Garey M.R., Johnson D.S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York, NYGoogle Scholar
- Grotschel, M. and Lovasz, L. (1995), Combinatorial optimization, In: Graham, R., Grotschel, M. and Lovasz, L. (eds.), Handbook of Combinatorics, vol. II. pp. 1541–1597. North-Holland, Amsterdam.Google Scholar
- Hoos H.H., Stutzle T. (2004), Stochastic Local Search: Foundations and Applications. Morgan Kaufmann, San FranciscoGoogle Scholar
- Schulz, A.S. and Weismantel, R. (1999), An oracle-polynomial time augmentation algorithm for integer programming, Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, MD, 967–968.Google Scholar
- Schulz A.S., Weismantel R., Ziegler G.M. (1995). 0/1-Integer programming: optimization and augmentation are equivalent. In: Spirakis P. (eds). Lecture Notes in Computer Science. 979. Springer, Berlin, Germany, pp. 473–483Google Scholar