The Complexity of Approximate

We study the approximability of the following NP-complete (in their feasibility recognition forms) number theoretic optimization problems: 1. x 2 Z n with minimum max 1in jxij satisfying P n i=1 xiai = gcd(a1; : : : ; an). We present a polynomial-time algorithm which approximates a minimum gcd set for a1; : : : ; an within a factor 1+ln n and prove that this algorithm is best possible in the sense that unless NP DTIME(n O(log log n)), there is no polynomial-time algorithm which approximates a minimum gcd set within a factor (1 ? o(1)) ln n. Concerning the second problem, we prove under the slightly stronger complexity theory assumption, NP 6 6 DTIME(n poly(log n)), that there is no polynomial-time algorithm which approximates a `1-minimum gcd multiplier within a factor 2 log 1? n , where is an arbitrary small positive constant. Complementary to this result, there exists a polynomial-time algorithm, which computes a gcd multiplier x 2 Z n for a1; : : : ; an 2 Zwith kxk1 0:5 kak1. In this paper, we also present a simple polynomial-time algorithm which computes a gcd multiplier x 2 Z n with Euclidean length kxk 1:5 n kak= gcd(a1; : : : ; an). Our inapproximability results rely on gap-preserving reductions from minimization problems with equal inapproximability ratios. We implicitly use the close connection between the hardness of approximation and the theory of interactive proof systems, particularly the work of 3, 9, 17, 14].