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 a ₁,..., a _n ∈ z, find a minimum gcd set for a ₁,..., a _n , i.e., a subset S $\subseteq${a ₁,..., a _n } with minimum cardinality satisfying gcd(S)=gcd(a ₁,..., a _n ).

• 2. Given n numbers a ₁,..., a _n ∈ z, find a ℓ_∞-minimum gcd multiplier for a ₁,..., a _n , i.e., a vector x ∈ zⁿ with minimum max_1≤i≤n ¦x_i¦ satisfying ∑ ⁿ_i=1 x _i a _i =gcd (a₁, ..., a _n ).

We present a polynomial-time algorithm which approximates a minimum gcd set for a ₁,..., a _n within a factor 1+ln n and prove that this algorithm is best possible in the sense that unless NP $\subseteq$ DTIME(n^{O(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 $\nsubseteq$ DTIME(n^{poly(log n)}), that there is no polynomial-time algorithm which approximates a ℓ_∞-minimum gcd multiplier within a factor $2^{\log ^{1 - \gamma } n}$, where γ is an arbitrary small positive constant.

Complementary to this result, there exists a polynomial-time algorithm, which computes a gcd multiplier x ∈ z ⁿ for a ₁,..., a_n ∈ z with ∥x∥_t8 ≤ 0.5 ∥a∥_t8. In this paper, we also present a simple polynomial-time algorithm which computes a gcd multiplier x ∈ z ⁿ with Euclidean length ∥x∥≤1.5ⁿ∥a∥/gcd(a ₁,..., a _n ).

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].

Supported by DFG under grant DFG-Leibniz-Programm Schn 143/5-1