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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.

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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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