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-@e, 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-@e; it is NP-hard to approximate the maximum number of simultaneously satisfiable equations of degree at most d to within d-@e for any @e; for any k>=4, it is NP-hard to approximate EQ"G[k] within any constant factor. These results extend Hastad's results (Hastad (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.

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