New Hardness Results for Diophantine Approximation

Abstract

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 α ∈ ℚⁿ, an error bound ε and a denominator bound N ∈ 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 ·α _i to its nearest integer is bounded by ε. 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 ∈ ℕ₊ such that the distances of Q·α _i to the nearest integer are bounded by ε is hard to approximate within a factor 2ⁿ 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.