Problem $2$: $3-$Approximation for Path$-$Constrained TSP

Theorem: There exists a $3-$approximation for the path$-$constrained traveling salesman problem.

Proof: We will call the edges and vertices of the constraint graph constraint edges and constraint vertices respectively.

First, take the complete graph on the n vertices with the distance d$($i$,$ j$)$ on the edge from i to j and form a new graph by adding an auxiliary vertex x with $0-$weight edges connected to each of the constraint vertices. Find a minimum spanning tree in this new graph. The optimal solution will contain the $0-$weight edges, plus a tree growing from each of the constraint vertices.

Discarding the $0-$weight edges and the auxiliary vertex x$,$ we have a forest where each tree contains exactly one of the constraint vertices. We produce a traveling salesman tour of each tree by the usual $$Twice$-$around$-$the$-$spanning$-$tree$$ technique. The sum of the tree costs is a lower bound on OPT because the optimal solution must have a path between consecutive constraint vertices, and removing the last edge on each of these paths produces a forest in which each tree contains exactly one of the constraint vertices.

Therefore, the sum of the costs of these tours is at most twice OPT. The final traveling salesman tour of the whole graph combines the individual tours with the $($undirected$)$ constraint edges, with shortcutting if necessary. The sum of all of the path edges is a lower bound on OPT by the triangle inequality, so the final solution is a $3-$approximation to the optimal path$-$constrained TSP$.$

Algorithm Design from Proof

Your task is to convert the above proof into a well$-$defined algorithm using pseudocode.

$1.$ Carefully analyze the provided proof to understand the key steps involved.

$2.$ Write the pseudocode for the algorithm, it means:

$$ Define the inputs and outputs clearly.

$$ Break down the proof into distinct algorithmic steps.

$$ Use control structures such as loops and conditionals where necessary.

$$ Do not add unnecessary details

$3.$ Analyze your algorithm:

$$ Explain why the algorithm achieves the $3-$approximation factor.

$$ Provide an analysis of the time complexity of the algorithm.