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, the \STC problem seeks a spanning tree with no tree-edge \emph{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 $\mathcal{O}(\sqrt{mn})$, 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 $\mathcal{O}(\sqrt{mn}\cdot \log n)$. We also present another algorithm for computing a spanning tree with congestion $\mathcal{O}(\sqrt{mn})$; this algorithm runs in sub-exponential time when $m = \omega(n \log^2 n)$. For achieving the above results, an important intermediate theorem is \emph{generalized Gy\H{o}ri-Lov{\'{a}}sz 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 $\mathcal{O}^*\left( 4^n \right)$, 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 \emph{expanding properties}, its STC is at most $\mathcal{O}(n)$, 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.