An Improved Quality Hierarchical Congestion Approximator in Near-Linear Time

2025-11-05

AI Summary

This paper addresses congestion prediction for single-commodity flows on graphs. We present the first near-linear-time (O(m polylog n)) algorithm for constructing a Hierarchical Congestion Approximator (HCA). Our method introduces the novel concept of *cut-edge routability*, designs a hierarchical construction mechanism that avoids recursive partitioning of large subgraphs, and integrates hierarchical cut families, an improved sparse-cut oracle, and parallelization techniques to handle multi-weighted vertices. The algorithm achieves an O(log²n log log n) congestion approximation ratio—the best known for any near-linear-time HCA. Its parallel variant exhibits near-linear work and O(polylog n) span, significantly enhancing practicality for efficient routing design and congestion management in distributed networks.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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. 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.

Problem

• Improving hierarchical congestion approximator construction speed and quality
• Reducing approximation gap between polynomial and near-linear algorithms
• Developing efficient parallel implementation with polylogarithmic span

Innovation

• Develops near-linear time hierarchical congestion approximator algorithm
• Introduces border routability concept for improved approximation quality
• Creates partitioning avoiding recursion on large subgraphs