An improved lower bound for approximating minimum GCD multiplier in l(infinity) norm (GCDM(infinity))
In this paper, we study the inapproximability of the following NP-complete number theoretic optimization problems introduced by Rossner and Seifert [C. Rossner, J.P. Seifert, The complexity of approximate optima for greatest common divisor computations, in: Proceedings of the 2nd International Algorithmic Number Theory Symposium, ANTS-II, 1996, pp. 307-322]: Given n numbers a(1),..., a(n) epsilon Z, find an l(infinity)-minimum GCD multiplier for a(1),..., a(n), i.e., a vector x epsilon Z(n) with minimum max 1   0 where n is the dimension of the given vector. This improves on the best previous result. The best result so far gave 2((logn)1-epsilon) factor hardness by Rossner and Seifert [C. Rossner, J.P. Seifert, The complexity of approximate optima for greatest common divisor computations, in: Proceedings of the 2nd international Algorithmic Number Theory Symposium, ANTS-II, 1996, pp. 307-322], where epsilon > 0 is an arbitrarily small constant. (C) 2007 Elsevier B.V. All rights reserved.
gcd approximation algorithm computational complexity np-hard number theoretic problems probabilistically checkable proofs