Fractional Covering with Upper Bounds on the Variables: Solving LPs with Negative Entries

We present a Lagrangian relaxation technique to solve a class of linear programs with negative coefficients in the objective function and the constraints. We apply this technique to solve (the dual of) covering linear programs with upper bounds on the variables: min {c ⊤ x|Ax ≥ b, x ≤ u, x ≥ 0} where \(c,u\in{\mathbb{R}}_{+}^m,b\in\mathbb{R}_+^n\) and \(A\in\mathbb{R}_+^{n\times m}\) have non-negative entries. We obtain a strictly feasible, (1+e)-approximate solution by making O(me − 2logm + min {n,loglogC}) calls to an oracle that finds the most-violated constraint. Here C is the largest entry in c or u, m is the number of variables, and n is the number of covering constraints. Our algorithm follows naturally from the algorithm for the fractional packing problem and improves the previous best bound of O(me − − 2log (mC)) given by Fleischer [1]. Also for a fixed e, if the number of covering constraints is polynomial, our algorithm makes a number of oracle calls that is strongly polynomial.