Chapter 10 The Traveling Salesman Problem

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10.2 Methods to solve the traveling salesman problem
10.2.1 Using the triangle inequality to solve the traveling salesman problem
Definition:
If for the set of vertices a, b, c ∈ V, it is true that t (a, c) ≤ t(a, b) + t(b, c) where t is the cost
function, we say that t satisfies the triangle inequality.
First, we create a minimum spanning tree the weight of which is a lower bound on the cost of an
optimal traveling salesman tour. Using this minimum spanning tree we will create a tour the cost
of which is at most 2 times the weight of the spanning tree. We present the algorithm that
performs these computations using the MST-Prim algorithm.
Approximation-TSP
Input: A complete graph G (V, E)
Output: A Hamiltonian cycle
1.select a “root” vertex r ∈ V [G].
2.use MST-Prim (G, c, r) to compute a minimum spanning tree from r.
3.assume L to be the sequence of vertices visited in a preorder tree walk of T.
4.return the Hamiltonian cycle H that visits the vertices in the order L.
The next set of figures show the working of the proposed algorithm.
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Figure 10.3 A set of cities and the resulting connection after the MST-Prim algorithm has been
applied..
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