Linear Programming with Two Variables Per Inequality in Poly-Log Time
George S. Lueker, Nimrod Megiddo, Vijaya Ramachandran · SIAM Journal on Computing · 1990
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 $O((\log m + \log^2 n) \log^2 n)$-time parallel algorithm for deciding feasibility is described. It requires $mn^{O(\log n)}$ 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 $O((\log m + \log ^2 n)^2 \log ^2 n)$ 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.