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Spanning Tree Congestion and Computation of Generalized Györi-Lovász Partition

Authors L. Sunil Chandran, Yun Kuen Cheung , Davis Issac

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Author Details

L. Sunil Chandran
  • Department of Computer Science and Automation, Indian Institute of Science, India
Yun Kuen Cheung
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Germany
Davis Issac
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Germany

Funding

  • Chandran, L. Sunil: This work was done while this author was visiting Max Planck Institute for Informatics, Saarbrücken, Germany, supported by Alexander von Humboldt Fellowship.
  • Cheung, Yun Kuen: Part of the work done while this author was a visitor at the Courant Institute, NYU. The visit was funded in part by New York University.

Cite AsGet BibTex

L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac. Spanning Tree Congestion and Computation of Generalized Györi-Lovász Partition. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.32

Abstract

We study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, it seeks a spanning tree with no tree-edge routing too many of the original edges. For any general connected graph with n vertices and m edges, we show that its STC is at most O(sqrt{mn}), which is asymptotically optimal since we also demonstrate graphs with STC at least Omega(sqrt{mn}). We present a polynomial-time algorithm which computes a spanning tree with congestion O(sqrt{mn}* log n). We also present another algorithm for computing a spanning tree with congestion O(sqrt{mn}); this algorithm runs in sub-exponential time when m = omega(n log^2 n). For achieving the above results, an important intermediate theorem is generalized Györi-Lovász theorem. Chen et al. [Jiangzhuo Chen et al., 2007] gave a non-constructive proof. We give the first elementary and constructive proof with a local search algorithm of running time O^*(4^n). We discuss some consequences of the theorem concerning graph partitioning, which might be of independent interest. We also show that for any graph which satisfies certain expanding properties, its STC is at most O(n), and a corresponding spanning tree can be computed in polynomial time. We then use this to show that a random graph has STC Theta(n) with high probability.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
Keywords
  • Spanning Tree Congestion
  • Graph Sparsification
  • Graph Partitioning
  • Min-Max Graph Partitioning
  • k-Vertex-Connected Graphs
  • Györi-Lovász Theorem

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References

  1. Ittai Abraham, Yair Bartal, and Ofer Neiman. Nearly tight low stretch spanning trees. In FOCS 2008, pages 781-790, 2008. Google Scholar
  2. Ittai Abraham and Ofer Neiman. Using petal-decompositions to build a low stretch spanning tree. In STOC 2012, pages 395-406, 2012. Google Scholar
  3. Noga Alon, Richard M. Karp, David Peleg, and Douglas B. West. A graph-theoretic game and its application to the k-server problem. SIAM J. Comput., 24(1):78-100, 1995. Google Scholar
  4. Nikhil Bansal, Uriel Feige, Robert Krauthgamer, Konstantin Makarychev, Viswanath Nagarajan, Joseph Naor, and Roy Schwartz. Min-max graph partitioning and small set expansion. SIAM J. Comput., 43(2):872-904, 2014. Google Scholar
  5. Sandeep N. Bhatt, Fan R. K. Chung, Frank Thomson Leighton, and Arnold L. Rosenberg. Optimal simulations of tree machines (preliminary version). In FOCS 1986, pages 274-282, 1986. Google Scholar
  6. Hans L. Bodlaender, Fedor V. Fomin, Petr A. Golovach, Yota Otachi, and Erik Jan van Leeuwen. Parameterized complexity of the spanning tree congestion problem. Algorithmica, 64(1):85-111, 2012. Google Scholar
  7. Hans L. Bodlaender, Kyohei Kozawa, Takayoshi Matsushima, and Yota Otachi. Spanning tree congestion of k-outerplanar graphs. Discrete Mathematics, 311(12):1040-1045, 2011. Google Scholar
  8. Jiangzhuo Chen, Robert D. Kleinberg, László Lovász, Rajmohan Rajaraman, Ravi Sundaram, and Adrian Vetta. (almost) tight bounds and existence theorems for single-commodity confluent flows. J. ACM, 54(4):16, 2007. Google Scholar
  9. Michael Elkin, Yuval Emek, Daniel A. Spielman, and Shang-Hua Teng. Lower-stretch spanning trees. SIAM J. Comput., 38(2):608-628, 2008. Google Scholar
  10. E. Győri. On division of graphs to connected subgraphs. Colloq. Math. Soc. Janos Bolyai, 18:485-494, 1976. Google Scholar
  11. Alexander Hoyer and Robin Thomas. The Győri-Lovász theorem. arXiv, abs/1605.01474, 2016. URL: http://arxiv.org/abs/1706.08115.
  12. David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. How easy is local search? J. Comput. Syst. Sci., 37(1):79-100, 1988. Google Scholar
  13. Marcos A. Kiwi, Daniel A. Spielman, and Shang-Hua Teng. Min-max-boundary domain decomposition. Theor. Comput. Sci., 261(2):253-266, 2001. Google Scholar
  14. Ioannis Koutis, Gary L. Miller, and Richard Peng. A nearly O(m log n) time solver for SDD linear systems. In FOCS 2011, pages 590-598, 2011. Google Scholar
  15. Kyohei Kozawa and Yota Otachi. Spanning tree congestion of rook’s graphs. Discussiones Mathematicae Graph Theory, 31(4):753-761, 2011. Google Scholar
  16. Kyohei Kozawa, Yota Otachi, and Koichi Yamazaki. On spanning tree congestion of graphs. Discrete Mathematics, 309(13):4215-4224, 2009. Google Scholar
  17. Hiu Fai Law, Siu Lam Leung, and Mikhail I. Ostrovskii. Spanning tree congestions of planar graphs. Involve, 7(2):205-226, 2014. Google Scholar
  18. László Lovász. A homology theory for spanning trees of a graph. Acta Math. Acad. Sci. Hungaricae, 30(3-4):241-251, 1977. Google Scholar
  19. Christian Löwenstein, Dieter Rautenbach, and Friedrich Regen. On spanning tree congestion. Discrete Math., 309(13):4653-4655, 2009. Google Scholar
  20. Yoshio Okamoto, Yota Otachi, Ryuhei Uehara, and Takeaki Uno. Hardness results and an exact exponential algorithm for the spanning tree congestion problem. J. Graph Algorithms Appl., 15(6):727-751, 2011. Google Scholar
  21. M. I. Ostrovskii. Minimal congestion trees. Discrete Math., 285:219-226, 2004. Google Scholar
  22. M. I. Ostrovskii. Minimum congestion spanning trees in planar graphs. Discrete Math., 310:1204-1209, 2010. Google Scholar
  23. M. I. Ostrovskii. Minimum congestion spanning trees in bipartite and random graphs. Acta Mathematica Scientia, 31(2):634-640, 2011. Google Scholar
  24. André Raspaud, Ondrej Sýkora, and Imrich Vrto. Congestion and dilation, similarities and differences: A survey. In SIROCCO 2000, pages 269-280, 2000. Google Scholar
  25. Shai Simonson. A variation on the min cut linear arrangement problem. Mathematical Systems Theory, 20(4):235-252, 1987. Google Scholar
  26. David Steurer. Tight bounds for the min-max boundary decomposition cost of weighted graphs. In SPAA 2006, pages 197-206, 2006. Google Scholar
  27. Hitoshi Suzuki, Naomi Takahashi, and Takao Nishizeki. A linear algorithm for bipartition of biconnected graphs. Inf. Process. Lett., 33(5):227-231, 1990. Google Scholar
  28. Zoya Svitkina and Éva Tardos. Min-max multiway cut. In APPROX-RANDOM 2004, pages 207-218, 2004. Google Scholar
  29. Koichi Wada and Kimio Kawaguchi. Efficient algorithms for tripartitioning triconnected graphs and 3-edge-connected graphs. In Graph-Theoretic Concepts in Computer Science, 19th International Workshop, WG '93, Utrecht, The Netherlands, June 16-18, 1993, Proceedings, pages 132-143, 1993. Google Scholar
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