Abstract
In parameterized complexity there are three natural definitions of fixed-parameter tractability called strongly uniform, weakly uniform and nonuniform fpt. Similarly, there are three notions of subexponential time, yielding three flavours of the exponential time hypothesis (ETH) stating that 3Sat is not solvable in subexponential time. It is known that ETH implies that p -Clique is not fixed-parameter tractable if both are taken to be strongly uniform or both are taken to be uniform, and we extend this to the nonuniform case. We also show that even the containment of weakly uniform subexponential time in nonuniform subexponential time is strict. Furthermore, we deduce from nonuniform ETH that no single exponent d allows for arbitrarily good fpt-approximations of clique.
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References
Downey, R., Fellows, M.: Fixed-parameter tractability and completeness iii: some structural aspects of the w hierarchy. In: Ambos-Spies, K., Homer, S., Schöning, U. (eds.) Complexity Theory, New York, NY, USA, pp. 191–225. Cambridge University Press (1993)
Ganian, R., Hlinený, P., Langer, A., Obdrzálek, J., Rossmanith, P., Sikdar, S.: Lower bounds on the complexity of MSO1 model-checking. In: Proc. STACS 2012, pp. 326–337 (2012)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? Journal of Computer and System Sciences 63, 512–530 (2001)
Chen, J., Huang, X., Kanj, I.A., Xia, G.: Linear fpt reductions and computational lower bounds. In: Proc. of STOC 2004, pp. 212–221 (2004)
Flum, J., Grohe, M.: Parametrized complexity and subexponential time (column: Computational complexity). Bulletin of the EATCS 84, 71–100 (2004)
Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54(1) (2007)
Downey, R.: The Birth and Early Years of Parameterized Complexity. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) Fellows Festschrift 2012. LNCS, vol. 7370, pp. 17–38. Springer, Heidelberg (2012)
Chen, Y., Flum, J.: On miniaturized problems in parameterized complexity theory. Theoretical Computer Science 351(3), 314–336 (2006)
Chen, Y., Grohe, M., Grüber, M.: On Parameterized Approximability. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 109–120. Springer, Heidelberg (2006)
Marx, D.: Parameterized complexity and approximation algorithms. The Computer Journal 51, 60–78 (2008)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
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Chen, Y., Eickmeyer, K., Flum, J. (2012). The Exponential Time Hypothesis and the Parameterized Clique Problem. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_4
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DOI: https://doi.org/10.1007/978-3-642-33293-7_4
Publisher Name: Springer, Berlin, Heidelberg
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