DERANDOMIZED GRAPH PRODUCTS

Berman and Schnitger gave a randomized reduction from approximating MAX-SNP problems within constant factors arbitrarily close to 1 to approximating clique within a factor of n(epsilon) (for some epsilon). This reduction was further studied by Blum, who gave it the name randomized graph products. We show that this reduction can be made deterministic (derandomized), using random walks on expander graphs. The main technical contribution of this paper is in proving a lower bound for 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 bounds for this probability.) This lower bound extends also to the case where different sets of vertices are specified for different time steps of the walk.