On the hardness of global and local approximation
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This paper combines local search and approximation into “approximation of local optima”, i.e., an attempt to escape hardness of exact local optimization by trying to find solutions which are approximately as good as the worst local optimum. The complexity of several well known optimization problems under this approach is investigated. Our main tool is a special reduction called the “strong PLS-reduction” which preserves cost and local search structure in a very strict sense. Completeness for the class of NP-optimization/local search problems under this reduction allows to deduce that both approximation of global and of local optima cannot be achieved efficiently (unless P = NP resp. P = PLS). We show that the (weighted) problems Min 4-DNF, MaxHopfield, Min/Max 0-1-Programming, Min Independent Dominating Set, and Min Traveling Salesman are all complete under the new reduction.
Moreover the unweighted Min 3-DNF problem is shown to be complete for the class of NP-optimization problems with polynomially bounded cost functions under an approximation preserving reduction. This implies that the logically defined class Min ∑0, the minimization analogue of max SNP, does not capture any (low) approximation degree.
KeywordsLocal Search Local Optimum Circuit Output Conjunction Vertex Polynomial Weight
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- [BeSc92]P. Berman, G. Schnitger. On the Complexity of Approximating the Independent Set Problem. Inform. and Comp., vol.96, pp. 77–94, 1992.Google Scholar
- [GJ79]M.R. Garey, D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979.Google Scholar
- [Ha93]M.M. Halldórson. Approximating the minimum maximal independence number. Inf. Proc. Lett., Vol.46, pp. 169–172, 1993.Google Scholar
- [HKP91]J. Hertz, A. Krogh, R.G. Palmer. Introduction to the Theory of Neural Computation. Addison-Wesley, 1991.Google Scholar
- [Ho82]J.J. Hopfield. Neural Networks and Physical Systems having Emergent Collective Computational Abilities, 1982. Reprinted in Anderson, Rosenfeld: Neurocomputing: Found, of Research, Cambridge, MIT-Press, 1988.Google Scholar
- [JPaY88]D.S. Johnson, C.H. Papadimitriou, M. Yannakakis. How Easy is Local Search? Journ. of Computer and System Sciences, vol.37, pp. 79–100, 1988.Google Scholar
- [Ka92]V. Kann. On the Approximability of NP-complete Optimization Problems. Dissertation, NADA Stockholm, 1992.Google Scholar
- [Ka93]V. Kann. Polynomially bounded minimization problems which are hard to approximate. Proc. 20th Int. Coll. on Automata, Lang, and Prog., pp. 52–63, 1993, Springer LNCS 700.Google Scholar
- [KMSV94]S. Khanna, R. Motwani, M. Sudhan, U. Vazirani. On Syntactic versus Computational Views of Approximability. 35th Symp. Found. Comput. Science, pp. 819–830, 1994.Google Scholar
- [KT94]P.G. Kolaitis, M.N. Thakur. Logical definability of NP optimization problems. Inform. and Comp., vol. 115, pp. 321–353, 1994.Google Scholar
- [Kr89]M.W. Krentel. Structure in locally optimal solutions. 30th Symp. Found. Comput. Science, pp. 216–221, 1989.Google Scholar
- [OM87]P. Orponen, H. Mannila. On approximation preserving reductions: Complete problems and robust measures. Technical Report C-1987-28, Department of Computer Science, University of Helsinki.Google Scholar
- [PapSY9O]C.H. Papadimitriou, A.A. Schäffer, M. Yannakakis. On the Complexity of Local Search. Proc. 22th ACM Symp. on Theory of Comp., pp. 438–445, 1990.Google Scholar
- [PapY91]C.H. Papadimitriou, M. Yannakakis. Optimization, Approximation, and Complexity Classes. Journ. of Computer and System Sciences, vol.43, pp. 425–440, 1991.Google Scholar
- [SchY91]A.A. Schäffer, M. Yannakakis. Simple Local Search Problems that are Hard to Solve. SIAM Journal Comput., vol.20, pp. 56–87, 1991.Google Scholar