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Abstract

Dimension with a fixed number of constraints can be computed with $O(s)$ basic arithmetic operations, where $s$ is the binary encoding length of the input. This improves on the quadratic running time of previous algorithms, which are based on Lenstra's algorithm and binary search. It follows that an integer program in fixed dimension, which is defined by $m$ constraints, each of binary encoding length at most $s$, can be solved with an expected number of $O(m + \log m \, s)$ arithmetic operations using Clarkson's random sampling algorithm.