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Integer plane multiflow maximisation: one-quarter-approximation and gaps

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Abstract

In this paper, we bound the integrality gap and the approximation ratio for maximum plane multiflow problems and deduce bounds on the flow-multicut-gap. We consider instances where the union of the supply and demand graphs is planar and prove that there exists a multiflow of value at least half the capacity of a minimum multicut. We then show how to convert any multiflow into a half-integer flow of value at least half the original multiflow. Finally, we round any half-integer multiflow into an integer multiflow, losing at most half the value thus providing a 1/4-approximation algorithm and integrality gap for maximum integer multiflows in the plane.

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Notes

  1. A reader familiar with the literature on multiflows with specified demands will note that the flow-cut gap usually refers to the ratio of the sparsest cut to the maximum fraction of demands that can be concurrently routed.

  2. Let \(G=C_4\) (a circuit on 4 vertices) and \(H=2K_2\) (complement of G) so \(G+H=K_4\). The supply edges have capacity 1 and the demands are capped at 1. An integer solution which routes 1 unit of each demand can only route flow half-integrally.

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Acknowledgements

Naveen Garg is the Janaki and K.A.Iyer Chair Professor at IIT Delhi.

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Authors and Affiliations

  1. Indian Institute of Technology Delhi, New Delhi, India

    Naveen Garg

  2. Hasso Plattner Institute, University of Potsdam, Potsdam, Germany

    Nikhil Kumar

  3. CNRS, Laboratoire G-SCOP, University of Grenoble Alpes, Grenoble, France

    András Sebő

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  1. Naveen Garg
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  3. András Sebő
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A preliminary version of this paper appeared in IPCO 20 [9]

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Garg, N., Kumar, N. & Sebő, A. Integer plane multiflow maximisation: one-quarter-approximation and gaps. Math. Program. 195, 403–419 (2022). https://doi.org/10.1007/s10107-021-01700-8

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