A first-order E-approximation algorithm for large linear programs

This report presents an algorithm that finds an -feasible solution relatively to some constraints of a linear program. The algorithm is a first-order feasible directions method with constant stepsize that attempts to find the minimizer of an exponential penalty function. When embedded with bisection search, the algorithm allows for the approximated solution of linear programs. The running time of our algorithm depends polynomially on 1/ and a parameter width introduced by Plotkin, Shmoys and Tardos in [3] and it is especially interesting when the direction finding (linear) subproblem is considered easy and amenable to reoptimization. We present applications of this framework to the Held and Karp bound on the traveling salesman problem and to a class of hard 0-1 linear programs. Computational results are expected to complement this report in the forthcoming revised version