Reducing the parallel complexity of certain linear programming problems

The parallel complexity of solving linear programming problems is studied in the context of interior point methods. If n and m, respectively, denote the number of variables and the number of constraints in the given problem, an algorithm that solves linear programming problems in O((mn)/sup 1/4/ (log 1 n)/sup 3/L) time using O(M(n)m/n+1n/sup 3/) processors is given. (M(n) is the number of operations for multiplying two n*n matrices). This gives an improvement in the parallel running time for n=o(m). A typical case in which n=o(m) is the dual of the uncapacitated transportation problem. The algorithm solves the uncapacitated transportation problem in O((VE)/sup 1/4/(log V)/sup 3/ (log V gamma )) time using O(V/sup 3/) processors, where V (E) is the number of nodes (edges) and gamma is the largest magnitude of an edge cost or a demand at a node. As a by-product, a better parallel algorithm for the assignment problem for graphs of moderate density is obtained.<>