Minimum Spanning Tree and Approximation of the Traveling Salesman Problem

A tree T is defined as a graph without any cycles. If edges are weighted, then we can define the minimum spanning tree (MST) of a graph G. The MST is the tree with smallest total edge weight that connects every node in G. Note that G must be a connected graph for any spanning tree, let alone a minimum spanning tree, to exist. We denote this MST as T ∗, which will be a subgraph of G (i.e., the edges of the graph T ∗ must be a subset of the edges of G). Finding the MST of a weighted graph feels like a combinatorial problem, the kind of thing that could easily be much more difficult than it would seem; however, we can use properties of the MST to find it easily using either of two different approaches. The key properties of MST that we will employ are as follows: