An Improved Quality Hierarchical Congestion Approximator in Near-Linear Time

A single-commodity congestion approximator for a graph is a compact data structure that approximately predicts the edge congestion required to route any set of single-commodity flow demands in a network. A hierarchical congestion approximator (HCA) consists of a laminar family of cuts in the graph and has numerous applications in approximating cut and flow problems in graphs, designing efficient routing schemes, and managing distributed networks. There is a tradeoff between the running time for computing an HCA and its approximation quality. The best polynomial-time construction in an $n$-node graph gives an HCA with approximation quality $O(\log^{1.5}n \log \log n)$. Among near-linear time algorithms, the best previous result achieves approximation quality $O(\log^4 n)$. We improve upon the latter result by giving the first near-linear time algorithm for computing an HCA with approximation quality $O(\log^2 n \log \log n)$. Additionally, our algorithm can be implemented in the parallel setting with polylogarithmic span and near-linear work, achieving the same approximation quality. This improves upon the best previous such algorithm, which has an $O(\log^9n)$ approximation quality. We also present a lower bound of $\Omega(\log n)$ for the approximation guarantee of hierarchical congestion approximators. Crucial for achieving a near-linear running time is a new partitioning routine that, unlike previous such routines, manages to avoid recursing on large subgraphs. To achieve the improved approximation quality, we introduce the new concept of border routability of a cut and provide an improved sparsest cut oracle for general vertex weights.