A Faster Algorithm for Two-Variable Integer Programming

Friedrich Eisenbrand, Sören Laue

Abstract: We show that a 2-variable integer program, defined by m constraints involving coefficients with at most ϕ bits can be solved with O(m + ϕ) arithmetic operations on rational numbers of size O(ϕ). 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 ϕ-bit integers and the problem of checking feasibility of an integer point for m constraints.