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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… (More)

We study the approximability of the following NP-complete (in their feasibility recognition forms) number theoretic optimization problems: 1. x 2 Z n with minimum max 1in jxij satisfying P n i=1 xiai = gcd(a1; : : : ; an). We present a polynomial-time algorithm which approximates a minimum gcd set for a1; : : : ; an within a factor 1+ln n and prove that… (More)

Given x 2 R n an integer relation for x is a non-trivial vector m 2 Z n with inner product hm; xi = 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)), thè 1-shortest integer relation for a given vector x 2 Q n cannot be approximated in polynomial time… (More)

We analyse a continued fraction algorithm (abbreviated CFA) for arbitrary dimension n showing that it produces simultaneous diophantine approximations which are up to the factor 2 (n+2)=4 best possible. Given a real vector x =(x1; : : : ; xn?1 ; 1) 2R n this CFA generates a sequence of vectors (p (k) jxi ? pi (k) =q (k) j 2 (n+2)=4 p 1 + x 2 i = jq (k) j 1+… (More)

We call a vector x 2 IR n highly regular if it satisses < x ; m > = 0 for some short, non{zero integer vector m where < : ; : > is the inner product. We present an algorithm which given x 2 IR n and 2 IN nds a highly regular nearby point x 0 and a short integer relation m for x 0 : The nearby point x 0 is 'good' in the sense that no short relation m of… (More)

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