The Complexity of the Extended GCD Problem

Abstract

We undertake a thorough complexity study of the following fundamental optimization problem, known as the ℓ_p-norm shortest extended GCD multiplier problem: given a ₁,..., a _n ∈ ℤ, find an ℓ_p-norm shortest gcd multiplier for a ₁,..., a _n, i.e., a vector x ∈ ℤⁿ with minimum $\left( {\sum\nolimits_{i = 1}^n {|x_i |^p } } \right)^{1/p}$ satisfying $\sum\nolimits_{i = 1}^n {x_i a_i = \gcd (a_1 ...,a_n )}$. First, we prove that the shortest GCD multiplier problem (in its feasibility recognition form) is NP-complete for every ℓ_p-norm with p ∈ ℤ. This gives an affirmative answer to a conjecture raised by Havas and Majewski. We then strengthen this negative result by ruling out even polynomial-time algorithms which only approximate an ℓ_p-norm shortest gcd multiplier within a factor $n^{1/(p\log ^\gamma n)}$ for γ an arbitrary small positive constant, under the widely accepted complexity theory assumption $NP \nsubseteq DTIME\left( {n^{poly(\log n)} } \right)$.

For positive results we focus on the ℓ₂-norm GCD multiplier problem. We show that approximating this problem within a factor of $\sqrt n$ is very unlikely NP-hard by placing it in NP ⋂ coAM through a simple constant-round interactive proof system. This result is complemented by a polynomial-time algorithm which computes an ℓ₂-norm shortest gcd multiplier up to a factor of 2^(n−1)/2.

This study is motivated by the importance of extended gcd calculations in applications in computational algebra and number theory. Our results rest upon the close connection between the hardness of approximation and the theory of interactive proof systems.

Partially supported by the Australian Research Council.

Partially supported by a UQ High Quality Research Grant.