A primal–dual interior point method whose running time depends only on the constraint matrix

We propose a primal–dual ‘layered-step’ interior point (LIP) algorithm for linear programming with data given by real numbers. This algorithm follows the central path, either with short steps or with a new type of step called a ‘layered least squares’ (LLS) step. The algorithm returns an exact optimum after a finite number of steps—in particular, after O(n^3.5c(A)) iterations, where c(A) is a function of the coefficient matrix. The LLS steps can be thought of as accelerating a classical path-following interior point method. One consequence of the new method is a new characterization of the central path: we show that it composed of at most n² alternating straight and curved segments. If the LIP algorithm is applied to integer data, we get as another corollary a new proof of a well-known theorem by Tardos that linear programming can be solved in strongly polynomial time provided that A contains small-integer entries.