We study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, it seeks a spanning tree with no tree-edge \emph{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 $O(\sqrt{mn}\cdot \log n)$.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 Gy\H{o}ri-Lov{\'{a}}sz theorem. Chen et al. in 2007 gave a non-constructive proof. We give the first elementary and constructive proof with a local search algorithm of running time $O^*\left( 4^n \right)$. 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. |