A non-iterative polynomial algorithm for linear programming

Abstract

Consider a linear programming problem with n primal and m dual variables paired with n dual and m primal slack variables respectively, and aggregately denote these variables and slack variables as a vector z of length 2(n+m). Unlike existing algorithms such as simplex and interior point methods solving linear programming by iteratively generating a sequence of feasible points to approach the optimal solution, the paper defines a function f mapping the constraint matrix and the right-hand side and objective vectors defining the linear programming problem to a binary vector of length n+m. It is shown that, under the uniqueness assumption of the optimal solution z* and for each (primal or dual) variable z_i, f_i is zero if and only if the optimal value z*_i is zero, and f_i is one if and only if z*_i is positive. Computation of f_i for each i consists of solving two groups of linear equations using O(m^2n) operations. Computing f_i is then non-iterative and independent of computing f_j for j <> i. Hence, at most O(m^2n^2) operations are required to compute f and consequently to solve the linear programming problem. The non-iterative and mutually independent features of computing the elements of f enable a parallel polynomial algorithm for linear programming.