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Bravely, moderately: a common theme in four recent works. (English) Zbl 1343.68113
Goldreich, Oded (ed.), Studies in complexity and cryptography. Miscellanea on the interplay between randomness and computation. In collaboration with Lidor Avigad, Mihir Bellare, Zvika Brakerski, Shafi Goldwasser, Shai Halevi, Tali Kaufman, Leonid Levin, Noam Nisan, Dana Ron, Madhu Sudan, Luca Trevisan, Salil Vadhan, Avi Wigderson, David Zuckerman. Berlin: Springer (ISBN 978-3-642-22669-4/pbk). Lecture Notes in Computer Science 6650, 373-389 (2011).
Summary: We highlight a common theme in four relatively recent works that establish remarkable results by an iterative approach. Starting from a trivial construct, each of these works applies an ingeniously designed sequence of iterations that yields the desired result, which is highly non-trivial. Furthermore, in each iteration, the construct is modified in a relatively moderate manner. The four works we refer to are
1.
the polynomial-time approximation of the permanent of non-negative matrices by M. Jerrum et al. [J. ACM 51, No. 4, 671–697 (2004; Zbl 1204.65044)];
2.
the iterative (Zig-Zag) construction of expander graphs by O. Reingold et al. [Ann. Math. (2) 155, No. 1, 157–187 (2002; Zbl 1008.05101)];
3.
the log-space algorithm for undirected connectivity by O. Reingold [“Undirected ST-connectivity in log-space”, in: Proceedings of the 37th annual ACM symposium on theory of computing, STOC 2005. New York, NY: ACM. 376–385 (2005; doi:10.1145/1060590.1060647)];
4.
and, the alternative proof of the PCP Theorem by I. Dinur [in: Proceedings of the 38th annual ACM symposium on theory of computing, STOC 2006. New York, NY: ACM Press. 241–250 (2006; Zbl 1301.68133)].

For the entire collection see [Zbl 1220.68005].

MSC:
68Q25 Analysis of algorithms and problem complexity
05C85 Graph algorithms (graph-theoretic aspects)
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
68W25 Approximation algorithms
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