Minimal $L^p$-congestion spanning trees on weighted graphs

TL;DR

This work extends spanning-tree congestion to the $L^p$ regime on weighted graphs by introducing $\mathcal{C}_p(G,\omega)$ and detailing its fundamental properties, including convergence to the classical congestion as $p\to\infty$ and stability under weight perturbations. It develops three general polynomial-time algorithms to approximate minimal $L^p$-STC in arbitrary weighted graphs and two planar-graph-specific variants, leveraging dual graphs and low-stretch insights. The paper provides lower bounds, exact results for complete and multipartite graphs, and extensive computational experiments on planar, nonplanar, and weighted graphs, showing that the Congestion Descent method often yields superior bounds while the planar methods offer fast heuristics. The results yield practical, “superoptimal” spanning trees that minimize multiple congestion criteria and improve known bounds for challenging classes like hypercubes, with implications for network design and robust routing under weighted constraints.

Abstract

A generalization of the notion of spanning tree congestion for weighted graphs is introduced. The $L^p$ congestion of a spanning tree is defined as the $L^p$ norm of the edge congestion of that tree. In this context, the classical congestion is the $L^\infty$-congestion. Explicit estimations of the minimal spanning tree $L^p$ congestion for some families of graphs are given. In addition, we introduce a polynomial-time algorithm for approximating the minimal $L^p$-congestion spanning tree in any weighted graph and another two similar algorithms for weighted planar graphs. The performance of these algorithms is tested in several graphs.

Figures (4)

• Figure 1: Example of a cycle perturbation in a spanning tree. Observe that only the edge congestion of bold edges can change.
• Figure 2: Radial dual trees do not need to be optimal for weighted planar graphs.
• Figure 3: Example of a switch at level $2$ in a BFS tree. Bold lines represent the tree at level $2$, dotted and dashed edges are those in $E^{T_2}$, and $T_e$ is obtained by switching the unique edge of $T_2$ adjacent to $u$ with $e$. If the tree lies in the dual graph, then dotted and dashed edges are dual to edges where the congestion can be computed. From $T_2$ to $T_e$ only the edge-congestion of the (duals of) dashed segments, which are those incident in $u$, may change.
• Figure 4: Some values of the estimated values of $\text{$\mathcal{C}$}_\infty(G(n,p))$ for the $\rm{sCD}_{\infty}$ algorithm, for $n\in\{10,15,20,\dots,100\}$ and $p=2\log(n)/n$. Observe that the $L^\infty$-congestion has a linear trend as expected.

Theorems & Definitions (63)

• Definition 2.1: edge congestion
• Definition 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Claim 2.5
• Proposition 2.6
• Proposition 2.7
• Proposition 2.8
• proof