Complexity of Linear Optimization Problems with Integer Data

Overview

The aim of complexity theory is to evaluate the minimal number of operations required to compute a solution of a given problem (and to determine the corresponding algorithm). This theory is far from being closed, since the answer is not known even for solving a linear system Ax = b, with A a matrix n × n invertible: classical factorization methods have an O(n ³) complexity but certain fast algorithms have an O(n ^α) complexity, with α < 2.5; the minimal value of α (proved to be larger than 2) is still unknown (see Coppersmith and Winograd [82]).