Simple and Fast Algorithms for Linear and Integer Programs With Two Variables per Inequality

The authors present an $O(mn^2 \log m)$ algorithm for solving feasibility in linear programs with up to two variables per inequality which is derived directly from the Fourier--Motzkin elimination method. (The number of variables and inequalities are denoted by $n$ and $m$, respectively.) The running time of the algorithm dominates that of the best known algorithm for the problem, and is far simpler. Integer programming on monotone inequalities, i.e., inequalities where the coefficients are of opposite sign, is then considered. This problem includes as a special case the simultaneous approximation of a rational vector with specified accuracy, which is known to be NP-complete. However, it is shown that both a feasible solution and an optimal solution with respect to an arbitrary objective function can be computed in pseudo-polynomial time.