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Maximum edge-disjoint paths in planar graphs with congestion 2

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Abstract

We study the maximum edge-disjoint path problem (medp) in planar graphs \(G=(V,E)\) with edge capacities u(e). We are given a set of terminal pairs \(s_it_i\), \(i=1,2 \ldots , k\) and wish to find a maximum routable subset of demands. That is, a subset of demands that can be connected by a family of paths that use each edge at most u(e) times. It is well-known that there is an integrality gap of \(\Omega (\sqrt{n})\) for the natural LP relaxation, even in planar graphs (Garg–Vazirani–Yannakakis). We show that if every edge has capacity at least 2, then the integrality gap drops to a constant. This result is tight also in a complexity-theoretic sense: recent results of Chuzhoy–Kim–Nimavat show that it is unlikely that there is any polytime-solvable LP formulation for medp which has a constant integrality gap for planar graphs. Along the way, we introduce the concept of rooted clustering which we believe is of independent interest.

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Notes

  1. One may relax the condition that a terminal’s demand is assigned to only one cluster. A splittable rooted clustering is one where a demand of \(d_i\) may be split across multiple clusters.

  2. If G is undirected, we bi-direct edges and “trim” t to be a sink.

  3. Their set-up is slightly different (e.g., they only consider bounded degree graphs) consider planar graphs and clusters being but their ideas easily yield this general result.

  4. This is essentially the same as one given in [7] as an example of a \(\Omega (\log n)\) gap for the congestion minimization LP for confluent flows.

  5. One could also consider splittable clusterings where a fraction of the demand is assigned across several clusters. We do not need to resort to this version.

  6. The notation \(\delta (X)\) represents the set of edges in the bipartite graph with exactly one endpoint in X.

  7. Informally, a contour is a closed simple curve in the plane; we refer the reader to texts in algebraic topology for more formality as needed.

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Acknowledgements

The authors are grateful for excellent feedback from three anonymous referees. The second author thanks Chandra Chekuri for many stimulating conversations on this and related topics. This work has been primarily supported by an NSERC Discovery Grant.

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  1. Cégep Édouard Montpetit, Longueuil, QC, Canada

    Loïc Séguin-Charbonneau

  2. Department of Computer Science, University of British Columbia, Vancouver, BC, Canada

    F. Bruce Shepherd

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  1. Loïc Séguin-Charbonneau
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  2. F. Bruce Shepherd
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Correspondence to F. Bruce Shepherd.

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Séguin-Charbonneau, L., Shepherd, F.B. Maximum edge-disjoint paths in planar graphs with congestion 2. Math. Program. 188, 295–317 (2021). https://doi.org/10.1007/s10107-020-01513-1

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