Approximation ratio on (1, 2)-metric Travelling Salesman Problem (TSP)

Question

I encountered a problem, where I am given a (fully-connected) graph within a metric space, where each edge weight is either 1 or 2.

My task is to prove that the following greedy algorithm gives a $\frac{3}{2}$-approximation for finding a TSP path: Start from an arbitrary node, pick the nearest neighbour node, go to this node and pick its nearest neighbour. Procedure is repeated until all nodes are picked, so that a valid cycle is produced.

I have seen more advanced techniques to solve the problem more efficiently here, which, however, did not help me.

My attempt was: Take any 3 nodes. Since we have a fully connected graph, the edges between the 3 nodes must form a triangle, and since we have a metric space, possible triangle sides are: $\{ (1,1,1), (2,2,2), (2,2,1) \}$. Since we are looking for path, we will never take all sides of this triangle, but at most 2. Since we always make our choice greedily, if the triangle is with sides $(2,2,1)$, we will never pick the 2 sides of length 2 together, but we will pick a side of length 1, and a side of length 2, hence resulting of weight from this triangle is 3.

I am stopping here, as I am not confident already.

Answer 1

Call an edge of weight $1$ a light edge.
Call a node incident to a light edge a light node.
Suppose the optimal travelling cycle contains $k$ light edges.

Consider those $k$ edges. Since these edges are part of a cycle, a node is incident to at most two of them. Since every edge is incident to two nodes, there are at least $k$ nodes that are incident to them. That is, there are at last $k$ light nodes.

For each light node, the greedy algorithm will choose at least one light edge incident to it unless it is the last node chosen by the greedy algorithm. So at least $(k-1)/2$ light edges will be selected by the greedy algorithm.

Let $n$ be the number of nodes in the graph. Since the number of edges in the travelling cycle is $n$, $k\le n$.
The weight of the optimal travelling cycle is $2n - k$.
The weight of the cycle selected by the greedy algorithm is at most $2n - (k-1)/2$.

$$(2n-(k-1)/2)- \frac32(2n-k)=\frac{k+1}2-n\le \frac{n+n}2-n=0. \quad\checkmark$$.