Inapproximability results for equations over finite groups

Abstract

An equation over a finite group G is an expression of form w 1w2. wκ = 1G, where each wi is a variable, an inverted variable, or a constant from G; such an equation is satisfiable if there is a setting of the variables to values in G so that the equality is realized. We study the problem of simultaneously satisfying a family of equations over a finite group G and show that it is NP-hard to approximate the number of simultaneously satisfiable equations to within |G|-ε for any ε > 0. This generalizes results of Håstad, who established similar bounds under the added condition that the group G is Abelian. © 2002 Springer-Verlag Berlin Heidelberg.