Multicommodity Flow in Trees: Packing via Covering and Iterated Relaxation
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- Könemann, J., Parekh, O. & Pritchard, D. Algorithmica (2014) 68: 776. doi:10.1007/s00453-012-9701-z
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We consider the max-weight integral multicommodity flow problem in trees. In this problem we are given an edge-, arc-, or vertex-capacitated tree and weighted pairs of terminals, and the objective is to find a max-weight integral flow between terminal pairs subject to the capacities. This problem is APX-hard and a 4-approximation for the edge- and arc-capacitated versions is known. Some special cases are exactly solvable in polynomial time, including when the graph is a path or a star.
We show that all three versions of this problems fit in a common framework: first, prove a counting lemma in order to use the iterated LP relaxation method; second, solve a covering problem to reduce the resulting infeasible solution back to feasibility without losing much weight. The result of the framework is a 1+O(1/μ)-approximation algorithm where μ denotes the minimum capacity, for all three versions. A complementary hardness result shows this is asymptotically best possible. For the covering analogue of multicommodity flow, we also show a 1+Θ(1/μ) approximability threshold with a similar framework.
When the tree is a spider (i.e. only one vertex has degree greater than 2), we give a polynomial-time exact algorithm and a polyhedral description of the convex hull of all feasible solutions. This holds more generally for instances we call root-or-radial.
A preliminary version of this work appeared in Könemann et al. (Proc. 6th Int. Workshop Approx. & Online Alg. (WAOA), pp. 1–14, 2008).