Inapproximability results for equations over infinite groups

An equation over a group G is an expression of form w 1 … w k = 1 G , where each w i is either a variable, an inverted variable, or a group constant and 1 G denotes the identity element; such an equation is satisfiable if there is a setting of the variables to values in G such that the equality is realized (Engebretsen et al. (2002) [10]). In this paper, we study the problem of simultaneously satisfying a family of equations over an infinite group G . Let EQ G [ k] denote the problem of determining the maximum number of simultaneously satisfiable equations in which each equation has occurrences of exactly k different variables. When G is an infinite cyclic group, we show that it is NP-hard to approximate EQ 1 G [3] to within 48 / 47 − ϵ , where EQ 1 G [3] denotes the special case of EQ G [3] in which a variable may only appear once in each equation; it is NP-hard to approximate EQ 1 G [2] to within 30 / 29 − ϵ ; it is NP-hard to approximate the maximum number of simultaneously satisfiable equations of degree at most d to within d − ϵ for any ϵ ; for any k ≥ 4 , it is NP-hard to approximate EQ G [ k] within any constant factor. These results extend Håstad’s results (Håstad (2001) [17]) and results of (Engebretsen et al. (2002) [10]), who established the inapproximability results for equations over finite Abelian groups and any finite groups respectively.