# 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*∈ z^{n}with minimum max_{1≤i≤n}¦x_{i}¦ satisfying ∑_{i=1}^{n}*x*_{ i }*a*_{ i }=gcd (a_{1}, ...,*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**∥≤1.5^{n}∥**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].

## Key Words

Approximation algorithm computational complexity gcd label cover**NP**-hard number theoretic problems probabilistically checkable proofs 2-prover 1-round interactive proof systems

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