On the hardness of approximating shortest integer relations among rational numbers

Abstract

Given x ϵ R n an integer relation for x is a non-trivial vector m ϵ Z n with inner product 〈 m,x〉 = 0 . In this paper we prove the following: Unless every NP language is recognizable in deterministic quasi-polynomial time, i.e., in time O( n poly ( log n)), the ℓ ∞-shortest integer relation for a given vector x ϵ Q n cannot be approximated in polynomial time within a factor of 2 log 0.5 − γ n , where γ is an arbitrarily small positive constant. This result is quasi-complementary to positive results derived from lattice basis reduction. A variant of the well-known L 3-algorithm approximates for a vector x ϵ Q n the ℓ 2-shortest integer relation within a factor of 2 n 2 in polynomial time. Our proof relies on recent advances in the theory of probabilistically checkable proofs, in particular on a reduction from 2-prover 1-round interactive proof-systems. The same inapproximability result is valid for finding the ℓ ∞-shortest integer solution for a homogeneous linear system of equations over Q.