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

TL;DR

This paper introduces a generalized $L^p$-congestion measure for spanning trees in weighted graphs, provides bounds for specific graph families, and develops polynomial-time algorithms for approximation, including for planar graphs.

Contribution

It generalizes spanning tree congestion to $L^p$ norms, offers explicit bounds, and presents efficient algorithms for approximating minimal $L^p$-congestion spanning trees.

Findings

Explicit bounds for $L^p$-congestion in certain graph families.

Polynomial-time approximation algorithms for general and planar graphs.

Empirical testing of algorithm performance on various graphs.

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.