A Network-Flow Technique for Finding Low-Weight Bounded-Degree Spanning Trees

TL;DR

This paper presents a network-flow heuristic for constructing low-weight, bounded-degree spanning trees in graphs with triangle inequality, providing approximation guarantees and counterexamples to previous conjectures.

Contribution

It introduces a network-flow based method for adjusting spanning trees to meet degree bounds and offers approximation algorithms with performance guarantees.

Findings

Approximation algorithms with performance guarantee less than 2 for geometric graphs.

Disproof of a conjecture relating TSP and MST costs in Euclidean graphs.

A heuristic method for degree-bounded spanning tree construction.

Abstract

The problem considered is the following. Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, compute a low-weight spanning tree such that the degree of each vertex is at most its specified bound. The problem is NP-hard (it generalizes Traveling Salesman (TSP)). This paper describes a network-flow heuristic for modifying a given tree T to meet the constraints. Choosing T to be a minimum spanning tree (MST) yields approximation algorithms with performance guarantee less than 2 for the problem on geometric graphs with L_p-norms. The paper also describes a Euclidean graph whose minimum TSP costs twice the MST, disproving a conjecture made in ``Low-Degree Spanning Trees of Small Weight'' (1996).