Spanning Tree Congestion and Computation of Generalized Györi-Lovász Partition
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.
Spanning Tree Congestion
Graph Sparsification
Graph Partitioning
Min-Max Graph Partitioning
k-Vertex-Connected Graphs
Györi-Lovász Theorem
Theory of computation~Sparsification and spanners
32:1-32:14
Regular Paper
https://arxiv.org/abs/1802.07632
L. Sunil
Chandran
L. Sunil Chandran
Department of Computer Science and Automation, Indian Institute of Science, India
This work was done while this author was visiting Max Planck Institute for Informatics, Saarbrücken, Germany, supported by Alexander von Humboldt Fellowship.
Yun Kuen
Cheung
Yun Kuen Cheung
Max Planck Institute for Informatics, Saarland Informatics Campus, Germany
https://orcid.org/0000-0002-9280-0149
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.
Davis
Issac
Davis Issac
Max Planck Institute for Informatics, Saarland Informatics Campus, Germany
https://orcid.org/0000-0001-5559-7471
10.4230/LIPIcs.ICALP.2018.32
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Sunil L. Chandran, Yun Kuen Cheung, and Davis Issac
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