Bravely, moderately: A common theme in four recent Works

Abstract

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 Jerrum, Sinclair, and Vigoda, 33rd STOC, 2001); 2 the iterative (Zig-Zag) construction of expander graphs (by Reingold, Vadhan, and Wigderson, 41st FOCS, 2000); 3 the log-space algorithm for undirected connectivity (by Reingold, 37th STOC, 2005); 4 and, the alternative proof of the PCP Theorem (by Dinur, 38th STOC, 2006). © 2011 Springer-Verlag Berlin Heidelberg.