Approximable minimization problems and optimal solutions on random inputs
In this paper we extend recent work about logical criteria for approximation properties of optimization problems. We focus on the relationship between logical expressibility and expected asymptotic growth of optimal solutions on random inputs. This further develops a probabilistic approach due to Behrendt, Compton and Grädel showing that expected optimal solutions for any problem in the class Max ⌆1 grows essentially like a polynomial. We show that there is a similar result for MinF+ II1, a syntactic class of minimization problems which provides a logical criterion for approximability. As a consequence, we show that some important problems do not belong to MinF+II1.
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- N. Alon and J. Spencer, The Probabilistic Method, Wiley, New York, 1991.Google Scholar
- Th. Behrendt, K. Compton and E. Grädel, Optimization Problems: Expressibility, Approximation Properties, and Expected Asymptotic Growth of Optimal Solutions, Computer Science Logic, 6th Workshop, CSL '92, San Miniato 1992, Selected Papers, vol. 702 of LNCS, Springer-Verlag, 1993, pp. 43–60.Google Scholar
- B. Bollobás, Random Graphs, Academic Press, London, 1985.Google Scholar
- K. Compton, 0-1 laws in logic and combinatorics, in NATO Adv. Study Inst. on Algorithms and Order, I. Rival, ed., D. Reidel, 1988, pp. 353–383.Google Scholar
- K. Compton and E. Grädel, Logical definability of counting functions, submitted for publication.Google Scholar
- R. Fagin, Generalized first-order spectra and polynomial time recognizable sets, in complexity of Computations, R. Karp, ed., vol. 7 of SIAM-AMS Proc., Providence, RI, 1974, American Math. Soc., pp. 43–73.Google Scholar
- Ph. Kolaitis and M. Thakur, Logical definability of NP optimization problems, Technical Report UCSC-CRL-90-48, Computer and Information Sciences, University of California, Santa Cruz (1990), to appear in Information and ComputationGoogle Scholar
- Ph. Kolaitis and M. Thakur, Approximation properties of NP minimization classes, Proceedings of 6th IEEE Conference on Structure in Complexity Theory (1991), 353–366, to appear in Journal of Computer and System Sciences.Google Scholar
- Ph. Kolaitis and M. Vardi, The decision problem for probabilities of higher order properties, in Proc. 19th ACM Symp. on Theory of Computing, New York, 1990, Association for Computing Machinery, pp. 446–456.Google Scholar
- C. Lautemann, Logical Definability of NP-Optimization Problems with Monadic Auxiliary Predicates, Computer Science Logic, 6th Workshop, CSL '92, San Miniato 1992, Selected Papers, vol. 702 of LNCS, Springer-Verlag, 1993, pp. 327–339.Google Scholar