Inapproximability results for equations over finite groups

Authors:
Lars Engebretsen;Jonas Holmerin;Alexander Russell
Affiliations:
Department of Numerical Analysis and Computer Science, Royal Institute of Technology, SE-100 44 Stockholm, Sweden;Department of Numerical Analysis and Computer Science, Royal Institute of Technology, SE-100 44 Stockholm, Sweden;Department of Computer Science and Engineering, University of Connecticut, Storrs, CT
Venue:
Theoretical Computer Science - Special issue on automata, languages and programming
Year:
2004

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Abstract

An equation over a finite group G is an expression of form w1w2...wk = 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 (J. ACM 48 (4) (2001) 798), who established similar bounds under the added condition that the group G is Abelian.