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An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs

Authors Nikolai Karpov, Marcin Pilipczuk, Anna Zych-Pawlewicz

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  • mimicking networks
  • planar graphs

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Abstract

Given an edge-weighted graph G with a set Q of k terminals, a mimicking network is a graph with the same set of terminals that exactly preserves the sizes of minimum cuts between any
partition of the terminals. A natural question in the area of graph compression is to provide as small mimicking networks as possible for input graph G being either an arbitrary graph or coming from a specific graph class.

In this note we show an exponential lower bound for cut mimicking networks in planar graphs: there are edge-weighted planar graphs with k terminals that require 2^(k-2) edges in any mimicking network. This nearly matches an upper bound of O(k * 2^(2k)) of Krauthgamer and Rika [SODA 2013, arXiv:1702.05951] and is in sharp contrast with the O(k^2) upper bound under the assumption that all terminals lie on a single face [Goranci, Henzinger, Peng, arXiv:1702.01136]. As a side result we show a hard instance for the double-exponential upper bounds given by Hagerup, Katajainen, Nishimura, and Ragde [JCSS 1998], Khan and Raghavendra [IPL 2014], and Chambers and Eppstein [JGAA 2013].

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Nikolai Karpov, Marcin Pilipczuk, and Anna Zych-Pawlewicz. An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 24:1-24:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.IPEC.2017.24

Author Details

Nikolai Karpov
Marcin Pilipczuk
Anna Zych-Pawlewicz

References

  1. Alexandr Andoni, Anupam Gupta, and Robert Krauthgamer. Towards (1 + ε)-approximate flow sparsifiers. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 279-293. SIAM, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.20.
  2. Erin W. Chambers and David Eppstein. Flows in one-crossing-minor-free graphs. J. Graph Algorithms Appl., 17(3):201-220, 2013. URL: http://dx.doi.org/10.7155/jgaa.00291.
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  5. Gramoz Goranci, Monika Henzinger, and Pan Peng. Improved guarantees for vertex sparsification in planar graphs. CoRR, abs/1702.01136, 2017. URL: http://arxiv.org/abs/1702.01136.
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