We give a strongly polynomial algorithm for minimum cost generalized flow and, as a consequence, for all linear programs with at most two nonzero entries per row, or at most two nonzero entries per column. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale's 9th problem.
Our approach is based on the recent ‘subspace layered least squares’ interior point method, an earlier joint work with Allamigeon, Dadush,
Loho and Natura. They show that the number of iterations needed by the IPM can be bounded in terms of the `straight line complexity’ of the central path. Roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. Our main contribution is a combinatorial analysis showing that the straight-line complexity of any minimum cost generalised flow instance is polynomial in the number of arcs and vertices.
This is joint work with Daniel Dadush, Zhuan Khye Koh, Bento Natura, and Neil Olver.
