Abstract
A new definition is given for the average growth of a function f: Σ * → IN with respect to a probability measure μ on Σ *. This allows us to define meaningful average case distributional complexity classes for arbitrary time bounds (previously, one could only distinguish between polynomial and superpolynomial growth). It is shown that basically only the ranking of the inputs by decreasing probabilities are of importance.
To compare the average and worst case complexity of problems we study average case complexity classes defined by a time bound and a bound on the complexity of possible distributions. Here, the complexity is measured by the time to compute the rank functions of the distributions. We obtain tight and optimal separation results between these average case classes. Also the worst case classes can be embedded into this hierarchy. They are shown to be identical to average case classes with respect to distributions of exponential complexity.
These ideas are finally applied to study the average case complexity of problems in NP. A reduction between distributional problems is defined for this new approach. We study the average case complexity class AvP consisting of those problems that can be solved by DTMs on the average in polynomial time for all distributions with efficiently computable rank function. Fast algorithms are known for some NP-complete problems under very simple distributions. For langugages in NP we consider the maximal allowable complexity of distributions such that the problem can still be solved efficiently by a DTM, at least on the average. As an example we can show that either the satisfiability problem remains hard, even for simple distributions, or NP is contained in AvP, that means every problem in NP can be solved efficiently on the average for arbitrary not too complex distributions.
Preview
Unable to display preview. Download preview PDF.
References
S. Ben-David, B. Chor, O. Goldreich, M. Luby, On the Theory of Average Case Complexity, J. CSS 44, 1992, 193–219; see also Proc. 21. STOC, 1989, 204–261.
Y. Gurevich, Average Case Completeness, J. CSS 42, 1991, 346–398.
R. Impagliazzo, L. Levin, No Better Ways to Generate Hard NP Instances than Picking Uniformly at Random, Proc. 31. FoCS, 1990, 812–821.
D. Johnson, The NP-Completeness Column, J. of Algorithms 5, 1984, 284–299.
L. Levin, Average Case Complete Problems, SIAM J. Computing 15, 1986, 285–286.
P. Miltersen, The Complexity of Malign Ensembles, Proc. 6. Structure in Complexity Theory, 1991, 164–171.
D. Mitchell, B. Selman, H. Levesque Hard and Easy Distributions of SAT Problems, Proc. 10. Nat. Conf. on Artificial Intelligence, 1992, 459–465.
R. Reischuk, Chr. Schindelhauer, Precise Average Case Complexity Measures, Technical Report, Technische Hochschule Darmstadt, 1992.
Chr. Schindelhauer, Neue Average Case Komplexitätsklassen, Diplomarbeit, Technische Hochschule Darmstadt, 1991.
R. Venkatesan, L. Levin, Random Instances of Graph Coloring Problems are Hard, Proc. 20. SToC, 1988, 217–222.
J. Wang, J. Belanger, On Average P vs. Average NP, Proc. 7. Struc. Compl., 1992.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Reischuk, R., Schindelhauer, C. (1993). Precise average case complexity. In: Enjalbert, P., Finkel, A., Wagner, K.W. (eds) STACS 93. STACS 1993. Lecture Notes in Computer Science, vol 665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56503-5_64
Download citation
DOI: https://doi.org/10.1007/3-540-56503-5_64
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56503-1
Online ISBN: 978-3-540-47574-3
eBook Packages: Springer Book Archive