Faster approximation algorithms for packing and covering problems

We adapt a method proposed by Nesterov [19] to design an algorithm that computes optimal solutions to fractional packing problems by solving O∗( −1 √ Kn) separable convex quadratic programs, where K is the maximum number of non-zeros per row and n is the number of variables. We also 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 -optimal flow by solving shortest path problems – the number of shortest paths computed grows as O( −1 log( )) in , and polynomially in the size of the problem. In contrast, previous algorithms required Ω( −2) iterations. We also describe extensions to other problems, such as the maximum multicommodity flow problem and covering problems.

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