Topic
Average-case complexity
About: Average-case complexity is a research topic. Over the lifetime, 1749 publications have been published within this topic receiving 44972 citations.
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TL;DR: In this article, a method based on Kolmogorov complexity is proposed to prove lower bounds on communication complexity, which is close to information theoretic methods and is applied to the hidden matching problem.
Abstract: We introduce a method based on Kolmogorov complexity to prove lower bounds on communication complexity. The intuition behind our technique is close to information theoretic methods. We use Kolmogorov complexity for three different things: first, to give a general lower bound in terms of Kolmogorov mutual information; second, to prove an alternative to Yao's minmax principle based on Kolmogorov complexity; and finally, to identify hard inputs. We show that our method implies the rectangle and corruption bounds, known to be closely related to the subdistribution bound. We apply our method to the hidden matching problem, a relation introduced to prove an exponential gap between quantum and classical communication. We then show that our method generalizes the VC dimension and shatter coefficient lower bounds. Finally, we compare one-way communication and simultaneous communication in the case of distributional communication complexity and improve the previous known result.
2 citations
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02 Mar 1995TL;DR: An average case measure is defined for the time complexity of circuits, using this notion tight upper and lower bounds could be obtained for the average case complexity of several basic Boolean functions.
Abstract: In contrast to machine models like Turing machines or random access machines, circuits are a rigid computational model. The internal information flow of a computation is fixed in advance, independent of the actual input. Therefore, in complexity theory only worst case complexity measures have been used to analyse this model. In [JRS94] we have defined an average case measure for the time complexity of circuits. Using this notion tight upper and lower bounds could be obtained for the average case complexity of several basic Boolean functions.
2 citations
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TL;DR: The duality theorem of integer programming and techniques to prove formula-size lower bounds as fundamental subjects in mathematical programming and computational complexity, respectively are explained.
Abstract: In this paper, we review subadditive approaches which arise in the theory of mathematical programming and computational complexity. In particular, we explain the duality theorem of integer programming and techniques to prove formula-size lower bounds as fundamental subjects in mathematical programming and computational complexity, respectively. We discuss parallel visions of these two different areas by showing some connections between them.
2 citations
01 Jan 2008
Abstract: The discussion of completeness in Section 3 follows the treatment of Goldreich’s notes [Gol97]. The proof that the Bounded Halting problem is complete for 〈NP,PComp〉 presented in Section 3.2 is due to Gurevich [Gur91]. After we present the 〈NP,PComp〉-completeness of Bounded Halting, we do not discuss further completeness results, of which a few are known, including the Tiling problem studied in Levin’s [Lev86] foundational paper, a graph coloring problem [VL88, LV18], a matrix decomposition problem [Gur90, BG95], a bounded version of the Post correspondence problem [Gur91], and diophantine matrix problems [VR92].
2 citations