 
Summary: Derandomized Graph Products
Noga Alon
Uriel Feige
Avi Wigderson
David Zuckerman§
February 22, 2002
Abstract
Berman and Schnitger [10] gave a randomized reduction from approximating MAX
SNP problems [24] within constant factors arbitrarily close to 1 to approximating clique
within a factor of n (for some ). This reduction was further studied by Blum [11],
who gave it the name randomized graph products. We show that this reduction can be
made deterministic (derandomized), using random walks on expander graphs [1]. The
main technical contribution of this paper is in lower bounding the probability that all
steps of a random walk stay within a specified set of vertices of a graph. (Previous work
was mainly concerned with upper bounding this probability.) This lower bound extends
also to the case that different sets of vertices are specified for different time steps of the
walk.
1 Introduction
We present lower bounds on the probability that all steps of a random walk stay within a
specified set of vertices of a graph. We then apply these lower bounds to amplify unapprox
