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)^1/4 (log 1 n)³L) time using O(M(n)m/n+1n³ ) 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)^1/4(log V)³ (log Vγ)) time using O(V³) processors, where V (E) is the number of nodes (edges) and γ 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