ANTS 1996: Algorithmic Number Theory pp 307-322

# The complexity of approximate optima for greatest common divisor computations

• Carsten Rössner
• Jean-Pierre Seifert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1122)

## Abstract

We study the approximability of the following NP-complete (in their feasibility recognition forms) number theoretic optimization problems:

1. 1.

Given n numbers a1,..., a n ∈ z, find a minimum gcd set for a1,..., a n , i.e., a subset S$$\subseteq$${a1,..., a n } with minimum cardinality satisfying gcd(S)=gcd(a1,..., a n ).

2. 2.

Given n numbers a1,..., a n z, find a ℓ-minimum gcd multiplier for a1,..., a n , i.e., a vector x ∈ zn with minimum max1≤in ¦xi¦ satisfying ∑ i=1 n x i a i =gcd (a1, ..., a n ).

We present a polynomial-time algorithm which approximates a minimum gcd set for a1,..., a n within a factor 1+ln n and prove that this algorithm is best possible in the sense that unless NP$$\subseteq$$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$$\nsubseteq$$DTIME(npoly(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 xzn for a1,..., anz with ∥xt8 ≤ 0.5 ∥at8. In this paper, we also present a simple polynomial-time algorithm which computes a gcd multiplier xzn with Euclidean length ∥x∥≤1.5na∥/gcd(a1,..., 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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