A Faster Algorithm for Two-variable Integer Programming

We show that a 2-variable integer program, defined by $m$ constraints involving coefficients with at most $\varphi$ bits can be solved with $O(m + \varphi)$ arithmetic operations on rational numbers of size~$O(\varphi)$. This result closes the gap between the running time of two-variable integer programming with the sum of the running times of the Euclidean algorithm on $\varphi$-bit integers and the problem of checking feasibility of an integer point for $m$~constraints