ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
Distributed algorithms for multicommodity flow problems via approximate steepest descent framework
ACM Transactions on Algorithms (TALG)
Content placement via the exponential potential function method
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
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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.