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Multicommodity Flow in Trees: Packing via Covering and Iterated Relaxation

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Abstract

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).

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Notes

  1. In detail, start by computing an optimal y, then route ⌊y d ⌋ units of each commodity d and reduce the capacities accordingly. The LP drop equals the profit of the routed flow, and the new LP has an optimum with \(y < {\bf 1}\).

  2. In our notation, x i is simply the member of X in the ith triple, so x i =x j does not imply i=j.

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Acknowledgements

We would like to thank Joseph Cheriyan, Jim Geelen, András Sebő, and Chaitanya Swamy for useful discussions, and the WAOA and Algorithmica reviewers for their helpful comments.

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Authors and Affiliations

  1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, Canada

    Jochen Könemann

  2. Sandia National Laboratories, Albuquerque, NM, USA

    Ojas Parekh

  3. Department of Computer Science, Princeton University, Princeton, NJ, USA

    David Pritchard

Authors
  1. Jochen Könemann
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  2. Ojas Parekh
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  3. David Pritchard
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Correspondence to David Pritchard.

Appendix: Approximation for Vertex-WMFT

Appendix: Approximation for Vertex-WMFT

Proposition 35

There is a 5-approximation algorithm for vertex-WMFT.

Proof

We assume the reader is familiar with the approach in [9]. Now redefine Binned Tree Colouring so that we must respect vertex capacities instead of edge capacities, and where we change the limit on the number of bins at a leaf vertex v to n v ≤c v .

Now we claim that [9, Thm. 2.2] holds with 5k colours instead of 4k. The main thing to check is that we can complete the partial colouring returned by the recursive call—checking these details pertains to the last paragraph of their proof. A demand edge e joining v i and v j cannot use

  • any colour already assigned within e’s bin at v i , of which there are less than 2k

  • any colour already assigned within e’s bin at v j , of which there are less than 2k

  • any colour which is already assigned to c v edges passing through v, of which there are less than k, since at most k⋅c v edges pass through v.

Hence one of the 5k colours is still available for colouring of e, as needed. □

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Könemann, J., Parekh, O. & Pritchard, D. Multicommodity Flow in Trees: Packing via Covering and Iterated Relaxation. Algorithmica 68, 776–804 (2014). https://doi.org/10.1007/s00453-012-9701-z

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