The complexity of approximate optima for greatest common divisor computations

Abstract

We study the approximability of the following NP-complete (in their feasibility recognition forms) number theoretic optimization problems: 1. Given n numbers a1,…,an ∈ z, find a minimum gcd set for a1,…, an, i.e., a subset S ⊆ {a1,…, an} with minimum cardinality satisfying gcd(S) = gcd (a1,…, an). 2. Given n numbers a1,…, an ∈ z, find a ℓ∞-minimum gcd multiplier for a1,…, an, i.e., a vector x ∈ zn with minimum max1≤i≤n|xi| satisfying (Formula Presented) 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(nO(log log n)) there is no polynomial-time algorithm which approximates a minimum gcd set within a factor (1 - o(1)) In n. Concerning the second problem, we prove under the slightly stronger complexity theory assumption, NP ⊈ DTIME(nPoly(log n)) that there is no polynomial-time algorithm which approximates a ℓ∞-minimum gcd multiplier within a factor 2log1-γ n, where γ is an arbitrary small positive constant. Complementary to this result, there exists a polynomial-time algorithm, which computes a gcd multiplier x ∈ zn for a1,…,an, ∈ z with ||x||∞ ≤ 0.5||a||∞. In this paper, we also present a simple polynomial-time algorithm which computes a gcd multiplier x ∈ zn with Euclidean length ||x|| < 1.5n ||a||/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, 8, 16, 13].