An improved lower bound for approximating minimum GCD multiplier in l∞ norm (GCDM∞)

  • Authors:

  • WenBin Chen;Jiangtao Meng;Dengpan Yin

  • Affiliations:

  • Department of Computer Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China;Department of Computer Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China;Department of Computer Science, Southern University, USA and AM College, Baton Rouge, LA, USA

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2008

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Abstract

In this paper, we study the inapproximability of the following NP-complete number theoretic optimization problems introduced by Rossner and Seifert [C. Rossner, J.P. Seifert, The complexity of approximate optima for greatest common divisor computations, in: Proceedings of the 2nd International Algorithmic Number Theory Symposium, ANTS-II, 1996, pp. 307-322]: Given n numbers a"1,...,a"n@?Z, find an @?"~-minimum GCD multiplier for a"1,...,a"n, i.e., a vector x@?Z^n with minimum max"1"@?"i"@?"n|x"i| satisfying @?"i"="1^nx"ia"i=gcd(a"1,...,a"n). We show that assuming PNP, it is NP-hard to approximate the Minimum GCD Multiplier in @?"~ norm (GCDM"~) within a factor n^c^/^l^o^g^l^o^g^n for some constant c0 where n is the dimension of the given vector. This improves on the best previous result. The best result so far gave 2^(^l^o^g^n^)^^^1^^^-^^^@e factor hardness by Rossner and Seifert [C. Rossner, J.P. Seifert, The complexity of approximate optima for greatest common divisor computations, in: Proceedings of the 2nd International Algorithmic Number Theory Symposium, ANTS-II, 1996, pp. 307-322], where @e0 is an arbitrarily small constant.