LINEAR-PROGRAMMING WITH 2 VARIABLES PER INEQUALITY IN POLY-LOG TIME

The parallel time complexity of the linear programming problem with at most two variables per inequality is discussed. Let n and m denote the number of variables and the number of inequalities, respectively, in a linear programming problem. It is assumed that all inequalities are weak. Under the concurrent-read-exclusive-write PRAM model, an O((log m + log2 n) log2 n)-time parallel algorithm for deciding feasibility is described. It requires mmO(log n) processors in the worst case, though it is not known whether this bound is tight. When the problem is feasible, a solution can be computed within the same complexity. Moreover, linear programming problems with at most two nonzero coefficients in the objective function can be solved in poly-log time on a similar number of processors. Consequently, all these problems can be solved sequentially with only O((log m + log2 n)2 log2 n) space. (These bounds assume that numbers take O(1) space, and arithmetic on them takes O(1) time; the problem can still be solved in poly-log space as a function of the input size even if a Turing machine model with rational input is used instead.) It is also shown that if the underlying graph has bounded tree-width and an underlying tree is given, then the feasibility problem is in the class NC.