The computational complexity of simultaneous diophantine approximation problems
SIAM Journal on Computing
Polynomial time algorithms for finding integer relations among real numbers
SIAM Journal on Computing
On the hardness of approximating shortest integer relations among rational numbers
Theoretical Computer Science
Approximating Good Simultaneous Diophantine Approximations Is Almost NP-Hard
MFCS '96 Proceedings of the 21st International Symposium on Mathematical Foundations of Computer Science
New Hardness Results for Diophantine Approximation
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
EDF-schedulability of synchronous periodic task systems is coNP-hard
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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In this paper, we show that assuming PNP, it is hard to approximate the Shortest Integer Relation in @?"~ norm (SIR"~) 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^^^/^^^2^^^-^^^@e factor hardness by Rossner and Seifert [C. Rossner, J.P. Seifert, On the hardness of approximating shortest integer relations among rational numbers, Theoret. Comput. Sci. 209 (1-2) (1998) 287-297], where @e0 is an arbitrarily small constant. By the improved result of SIR"~, we also improve on the inapproximability factor of Good Diophantine Approximations in @?"~ norm (GDA"~) to n^c^/^l^o^g^l^o^g^n.