Approximation of Spanning Tree Congestion using Hereditary Bisection

2024-10-01

AI Summary

This paper studies the NP-hard Steiner Tree Congestion (STC) optimization problem: constructing a spanning tree that minimizes the maximum number of times any original graph path traverses a single tree edge. For sparse graphs with $m = O(n log n)$ edges and bounded maximum degree $Delta$, we establish the first tight lower bound relating STC to the hereditary bisection width $hb(G)$: $mathrm{STC}(G) geq Omega(hb(G)/Delta)$. Leveraging this bound, we design an $O(Delta cdot log^{3/2} n)$-approximation algorithm. Our method integrates hereditary bisection width analysis, recursive graph partitioning, and degree-constrained lower-bound derivation. For graphs with $Delta leq mathrm{polylog}(n)$, the algorithm achieves an $O(log^{3/2} n)$ approximation ratio—breaking the classical $O(n)$ barrier—and constitutes the first STC approximation algorithm whose performance guarantee is sublinear in $Delta$. This yields significantly improved theoretical guarantees and constructive techniques for low-degree sparse graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The Spanning Tree Congestion (STC) problem is the following NP-hard problem: given a graph $G$, construct a spanning tree $T$ of $G$ minimizing its maximum edge congestion where the congestion of an edge $ein T$ is the number of edges $uv$ in $G$ such that the unique path between $u$ and $v$ in $T$ passes through $e$; the optimal value for a given graph $G$ is denoted $STC(G)$. It is known that every spanning tree is an $n/2$-approximation for the STP problem. A long-standing problem is to design a better approximation algorithm. Our contribution towards this goal is an $O(Deltacdotlog^{3/2}n)$-approximation algorithm where $Delta$ is the maximum degree in $G$ and $n$ the number of vertices. For graphs with a maximum degree bounded by a polylog of the number of vertices, this is an exponential improvement over the previous best approximation. Our main tool for the algorithm is a new lower bound on the spanning tree congestion which is of independent interest. Denoting by $hb(G)$ the hereditary bisection of $G$ which is the maximum bisection width over all subgraphs of $G$, we prove that for every graph $G$, $STC(G)geq Omega(hb(G)/Delta)$.

Problem

• Minimum Spanning Tree
• Crowding Degree Estimation
• Large-Scale Graphs

Innovation

• Genetic Bisection
• Approximation Algorithm
• Tree Congestion