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Average-Case Non-approximability of Optimisation Problems

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Fundamentals of Computation Theory (FCT 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3623))

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Abstract

Both average-case complexity and the study of the approximability of NP optimisation problems are well established and active fields of research. Many results concerning the average behaviour of approximation algorithms for NP optimisation problems exist, both with respect to their running time and their performance ratio, but a theoretical framework to examine their structural properties with respect to their average-case approximability is not yet established. With this paper, we hope to fill the gap and provide not only the necessary definitions, but show that

  1. 1

    The class of NP optimisation problems with p-computable input distributions has complete problems with respect to an average-approximability preserving reduction.

  2. 2

    The average-time variants of worst-case approximation classes form a strict hierarchy if NP is not easy on average. By linking average-ratio approximation algorithms to p-time algorithms schemes, we can prove similar hierarchy results for the average-ratio versions of the approximation classes. This is done under the premise that not all NP-problems with p-computable input distributions have p-time algorithm schemes.

  3. 3

    The question whether NP is easy on average is equivalent to the question whether every NP optimisation problem with a p-computable input distribution has an average p-time approximation scheme.

Classification: Computational and structural complexity.

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Author information

Authors and Affiliations

  1. Fakultät IV – Elektrotechnik und Informatik, Technische Universität Berlin, 10587, Berlin, Germany

    Birgit Schelm

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  1. Birgit Schelm
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Editors and Affiliations

  1. Institut für Theoretische Informatik, Universität zu Lübeck, Germany

    Maciej Liśkiewicz

  2. Institut für Theoretische Informatik, Universität zu Lübeck, Ratzeburger Allee 160, 23538, Lübeck, Germany

    Rüdiger Reischuk

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Schelm, B. (2005). Average-Case Non-approximability of Optimisation Problems. In: Liśkiewicz, M., Reischuk, R. (eds) Fundamentals of Computation Theory. FCT 2005. Lecture Notes in Computer Science, vol 3623. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537311_36

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  • DOI: https://doi.org/10.1007/11537311_36

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28193-1

  • Online ISBN: 978-3-540-31873-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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