The computational complexity of simultaneous diophantine approximation problems
SIAM Journal on Computing
Polynomial-Time Aggregation of Integer Programming Problems
Journal of the ACM (JACM)
Approximating Good Simultaneous Diophantine Approximations Is Almost NP-Hard
MFCS '96 Proceedings of the 21st International Symposium on Mathematical Foundations of Computer Science
The Constrained Minimum Spanning Tree Problem (Extended Abstract)
SWAT '96 Proceedings of the 5th Scandinavian Workshop on Algorithm Theory
Approximating SVPinfty to within Almost-Polynomial Factors Is NP-Hard
CIAC '00 Proceedings of the 4th Italian Conference on Algorithms and Complexity
Production Planning by Mixed Integer Programming (Springer Series in Operations Research and Financial Engineering)
An improved lower bound for approximating Shortest Integer Relation in l∞ norm (SIR∞)
Information Processing Letters
SIAM Journal on Discrete Mathematics
The mixing-MIR set with divisible capacities
Mathematical Programming: Series A and B
Static-Priority Real-Time Scheduling: Response Time Computation Is NP-Hard
RTSS '08 Proceedings of the 2008 Real-Time Systems Symposium
The mixing set with divisible capacities
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
EDF-schedulability of synchronous periodic task systems is coNP-hard
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Mixing Sets Linked by Bidirected Paths
SIAM Journal on Optimization
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We revisit simultaneous Diophantine approximation , a classical problem from the geometry of numbers which has many applications in algorithms and complexity. The input to the decision version of this problem consists of a rational vector *** *** *** n , an error bound *** and a denominator bound N *** *** + . One has to decide whether there exists an integer, called the denominator Q with 1 ≤ Q ≤ N such that the distance of each number Q ·*** i to its nearest integer is bounded by *** . Lagarias has shown that this problem is NP-complete and optimization versions have been shown to be hard to approximate within a factor n c / loglogn for some constant c 0. We strengthen the existing hardness results and show that the optimization problem of finding the smallest denominator Q *** *** + such that the distances of Q ·*** i to the nearest integer are bounded by *** is hard to approximate within a factor 2 n unless ${\textrm{P}} = {\rm NP}$. We then outline two further applications of this strengthening: We show that a directed version of Diophantine approximation is also hard to approximate. Furthermore we prove that the mixing set problem with arbitrary capacities is NP-hard. This solves an open problem raised by Conforti, Di Summa and Wolsey.