The complexity and approximability of finding maximum feasible subsystems of linear relations
Theoretical Computer Science
Approximate solution of NP optimization problems
Theoretical Computer Science
Probabilistic checking of proofs and hardness of approximation problems
Probabilistic checking of proofs and hardness of approximation problems
Interactive proofs and the hardness of approximating cliques
Journal of the ACM (JACM)
The hardness of approximate optima in lattices, codes, and systems of linear equations
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
SIAM Journal on Computing
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
A PCP characterization of NP with optimal amortized query complexity
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Some optimal inapproximability results
Journal of the ACM (JACM)
The importance of being biased
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Inapproximability Results for Equations over Finite Groups
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
The Complexity of Solving Equations Over Finite Groups
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Hardness of approximate hypergraph coloring
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Clique is hard to approximate within n1-
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
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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.