The Complexity of the Extended GCD Problem

• Authors: George Havas;Jean-Pierre Seifert
• Venue: MFCS '99 Proceedings of the 24th International Symposium on Mathematical Foundations of Computer Science
• Year: 1999

We undertake a thorough complexity study of the following fundamental optimization problem, known as the lp-norm shortest extended GCD multiplier problem: given a1,...,an Z, find an lp-norm shortest gcd multiplier for a1,...,an, i.e., a vector x ∈ Zn with minimum (Σi-1n|xi|p)1/p satisfying Σi=1nxiai = gcd(a1,...,an). First, we prove that the shortest GCD multiplier problem (in its feasibility recognition form) is NP-complete for every ip-noim with p ∈ N. This gives an affirmative answer to a conjecture raised by Haves and Majewski. We then strengthen this negative result by ruling out even polynomial-time algorithms which only approximate an lp-norm shortest gcd multiplier within a factor n1/p(plogλn), for λ an arbitrary small positive constant, under the widely accepted complexity theory assump- tion NP ⊈ DTIME(nPoly(log n)). For positive results we focus on the l2-norm GCD multiplier problem. We show that approximating this problem within a factor of √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-2)/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.