Approximating SVP∞ to within almost-polynomial factors is NP-hard

Approximating Svp 1 to within Almost-polynomial Factors Is Np-hard

This paper shows SVP 1 and CVP 1 to be NP-hard to approximate to within any factor up to n 1=loglogn. This improves on the best previous result ABSS93] that showed quasi-NP-hardness for smaller factors, namely 2 log 1?" n for any constant " > 0. We show a direct reduction from SAT to these problems, that combines ideas from ABSS93] and from DKS98, DKRS99], along with some modiications. Our resu...

Approximating-CVP to Within Almost-Polynomial Factors is NP-Hard

This paper shows the closest vector in a lattice to be NPhard to approximate to within any factor up to 2(logn)1 where = (log logn) c for any constant c < 12 . Introduction A lattice L = L(v1; ::; vn), for vectors v1; ::; vn 2 Rn is the set of all integer linear combinations of v1; ::; vn, that is, L = fP aivi j ai 2 Zg. Given a lattice L and an arbitrary vector y, the Closest Vector Problem (C...

Approximating Good Simultaneous Diophantine Approximations Is Almost NP-Hard

Given a real vector =(1; : : : ; d) and a real number " > 0 a good Diophantine approximation to is a number Q such that kQQ mod Zk1 ", where k k1 denotes thè1-norm kxk1 := max 1id jxij for x = (x1; : : : ; x d). Lagarias 12] proved the NP-completeness of the corresponding decision problem, i.e., given a vector 2 Q d , a rational number " > 0 and a number N 2 N+, decide whether there exists a nu...

Approximating the Svp to within a Factor ?

Approximating MAX-E3LIN is NP-Hard

This lecture focuses on the MAX-E3LIN problem. We prove that approximating it is NP-hard by a reduction from LABEL-COVER. In the MAX-E3LIN problem, our input is a series of linear equations (mod 2) in n binary variables, each with three terms. Equivalently, one can think of this as ±1 variables and ternary products. The objective is to maximize the fraction of satisfied equations.