Approximating fractional packings and coverings in O(1/ε) iterations

We adapt a method proposed by Nesterov [Math. Program. Ser. A, 103 (2005), pp. 127-152] to design an algorithm that computes epsilon-optimal solutions to fractional packing problems by solving O(epsilon(-1) root Knln(m)) separable convex quadratic programs, where n is the number of variables, m is the number of constraints, and K is the maximum number of nonzero elements in any constraint. We show that the quadratic program can be approximated to any degree of accuracy by an appropriately defined piecewise-linear program. For the special case of the maximum concurrent flow problem on a graph G = (V, E) with rational capacities and demands, we obtain an algorithm that computes an epsilon-optimal flow by solving shortest path problems, i.e., problems in which the number of shortest paths computed grows as O(epsilon(-1) log(epsilon(-1))) in epsilon and polynomially in the size of the problem. In contrast, previous algorithms required Omega(epsilon(-2)) iterations. We also describe extensions to the maximum multicommodity low problem, the pure covering problem, and mixed packing-covering problem.