DeepAI

# Spanning Tree Congestion and Computation of Generalized Győri-Lovász Partition

We study a natural problem in graph sparsification, the Spanning Tree Congestion () problem. Informally, the problem seeks a spanning tree with no tree-edge routing too many of the original edges. The root of this problem dates back to at least 30 years ago, motivated by applications in network design, parallel computing and circuit design. Variants of the problem have also seen algorithmic applications as a preprocessing step of several important graph algorithms. For any general connected graph with n vertices and m edges, we show that its STC is at most O(√(mn)), which is asymptotically optimal since we also demonstrate graphs with STC at least Ω(√(mn)). We present a polynomial-time algorithm which computes a spanning tree with congestion O(√(mn)· n). We also present another algorithm for computing a spanning tree with congestion O(√(mn)); this algorithm runs in sub-exponential time when m = ω(n ^2 n). For achieving the above results, an important intermediate theorem is generalized Győri-Lovász theorem, for which Chen et al. gave a non-constructive proof. We give the first elementary and constructive proof by providing a local search algorithm with running time O^*( 4^n ), which is a key ingredient of the above-mentioned sub-exponential time algorithm. We discuss a few 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 Θ(n) with high probability.

04/17/2020

### Low-stretch spanning trees of graphs with bounded width

We study the problem of low-stretch spanning trees in graphs of bounded ...
06/12/2018

### Drawing a Rooted Tree as a Rooted y-Monotone Minimum Spanning Tree

Given a rooted point set P, the rooted y-Monotone Minimum Spanning Tree ...
11/20/2019

### A 2-approximation for the k-prize-collecting Steiner tree problem

We consider the k-prize-collecting Steiner tree problem. An instance is ...
09/17/2022

### Better Hardness Results for the Minimum Spanning Tree Congestion Problem

In the spanning tree congestion problem, given a connected graph G, the ...
04/10/2019

### Minimum Spanning Trees in Weakly Dynamic Graphs

In this paper, we study weakly dynamic undirected graphs, that can be us...
05/20/2020

### A Unifying Model for Locally Constrained Spanning Tree Problems

Given a graph G and a digraph D whose vertices are the edges of G, we in...
10/23/2020

### Quickly excluding a non-planar graph

A cornerstone theorem in the Graph Minors series of Robertson and Seymou...