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by Carsten Rossner, Jean-pierre Seifert
http://www.informatik.uni-frankfurt.de/~roessner/group/roessneren/papers/mfcs96.ps
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Abstract:
Abstract. Given a real vector ff=(ff1; : : : ; ff d) and a real number " ? 0 a good Diophantine approximation to ff is a number Q such that kQff mod Zk1 ", where k \Delta k1 denotes the `1-norm kxk1: = max 1id jx i j for x = (x1; : : : ; xd). Lagarias [12] proved the NP-completeness of the corresponding decision problem, i.e., given a vector ff 2 Q d, a rational number " ? 0 and a number N 2 N+, decide whether there exists a number Q with 1 Q N and kQff mod Zk1 ". We prove that, unless NP ` DTIME(n poly(log n)), there exists no polynomial-time algorithm which computes on inputs ff 2 Q
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