Approximating Fractional Packings and Coverings in O(1/epsilon) Iterations

Abstract

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 $O(\epsilon^{-1}\sqrt{Kn\ln(m)})$ 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 flow problem, the pure covering problem, and mixed packing-covering problem.