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Given x small epsilon, Greek Rn an integer relation for x is a non-trivial vector m small epsilon, Greek Zn 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(npoly(log n)), the ℓinfinity-shortest integer relation for a given vector x small epsilon, Greek Qn cannot be approximated in polynomial time within a factor of 2log0.5 − small gamma, Greekn, where small gamma, Greek 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 L3-algorithm approximates for a vector x small epsilon, Greek Qn the ℓ2-shortest integer relation within a factor of 2n/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 ℓinfinity-shortest integer solution for a homogeneous linear system of equations over Q.
We study the approximability of the following NP-complete (in their feasibility recognition forms) number theoretic optimization problems: 1. Given n numbers a1 ; : : : ; an 2 Z, find a minimum gcd set for a1 ; : : : ; an , i.e., a subset S fa1 ; : : : ; ang with minimum cardinality satisfying gcd(S) = gcd(a1 ; : : : ; an ). 2. Given n numbers a1 ; : : : ; an 2 Z, find a 1-minimum gcd multiplier for a1 ; : : : ; an , i.e., a vector x 2 Z n with minimum max 1in jx i j satisfying P n...