On the advantage over a random assignment

Johan Håstad, Srinivasan Venkatesh

Abstract: We initiate the study of a new measure of approximation. This measure compares the performance of an approximation algorithm to the random assignment algorithm. Since the random assignment algorithm is known to give essentially the best possible polynomial time approximation algorithm for many optimization problems, this is a useful measure.In this paper, we focus on this measure for the optimization problems, Max-Lin-2, in which we need to maximize the number of satisfied linear equations in a system of linear equations modulo 2, and Max-$k$-Lin-2, a special case of the above problem in which each equation has at most $k$ variables. The main techniques we use, in our approximation algorithms and inapproximability results for this measure, are from Fourier analysis and derandomization.