In this chapter, we will study the spanning tree problem in undirected graphs. First, we will study an exact linear programming formulation and show its integrality using the iterative method. To do this, we will introduce the uncrossing method, which is a very powerful technique in combinatorial optimization. The uncrossing method will play a crucial role in the proof and will occur at numerous places in later chapters. We will show two different iterative algorithms for the spanning tree problem, each using a different choice of 1-elements to pick in the solution. For the second iterative algorithm, we show three different correctness proofs for the existence of a 1-element in an extreme point solution: a global counting argument, a local integral token counting argument and a local fractional token counting argument. These token counting arguments will be used in many proofs in later chapters.

We then address the degree-bounded minimum-cost spanning tree problem. We show how the methods developed for the exact characterization of the spanning tree polyhedron are useful in designing approximation algorithms for this NP-hard problem. We give two additive approximation algorithm: The first follows the first approach for spanning trees and naturally generalizes to give a simple proof of the additive two approximation result of Goemans [59]; the second follows the second approach for spanning trees and uses the local fractional token counting argument to provide a very simple proof of the additive one approximation result of Singh and Lau [125].