New Hardness Results for Diophantine Approximation

Research Area: Algorithm Engineering for Real-time Scheduling and Routing Year: 2009
Type of Publication: In Proceedings Keywords: Diophantine approximation; NP-hardness
Authors: F. Eisenbrand; T. Rothvoß
Volume: 5687
Book title: Proceedings of the 12th Intl. Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2009)
Series: Lecture Notes in Computer Science Pages: 98-110
ISBN: 978-3-642-03684-2
We revisit simultaneous diophantine approximation, a classical problem from the geometry of numbers which has many applications in algorithms and complexity. The input of the decision version of this problem consists of a rational vector alpha, an error bound epsilon 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*alpha_i to its nearest integer is bounded by epsilon. 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/ log log n 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*alpha_i to the nearest integer is bounded by epsilon is hard to aproximate within a factor 2^n unless P=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.
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