Speeding-up linear programming using fast matrix multiplication

The author presents an algorithm for solving linear programming problems that requires O((m+n)^1.5 nL) arithmetic operations in the worst case, where m is the number of constraints, n the number of variables, and L a parameter defined in the paper. This result improves on the best known time complexity for linear programming by about √n . A key ingredient in obtaining the speedup is a proper combination and balancing of precomputation of certain matrices by fast matrix multiplication and low-rank incremental updating of inverses of other matrices. Specializing the algorithm to problems such as minimum-cost flow, flow with losses and gains, and multicommodity flow leads to algorithms whose time complexity closely matches or is better than the time complexity of the best known algorithms for these problems