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Theorem 34.14: TSP $\in$ NPC

Variant of proof in textbook.

Proof sketch:

1.
TSP $\in$ NP
Given a tour, check that each vertex is visited exactly once and the sum of costs $\leq$ k

2.
L' = HC

3.
HC \(\leq_P\) TSP
Given graph G = (V, E), transformation f outputs complete graph with vertices V.
Weights of edges = 1 if \(e \in E\), or (|V| + 1) if \(e \not\in E\)
Also outputs the number |V|.
f is clearly implementable in polynomial time.

4.
Then there exists a tour in this complete graph of size \(\leq \vert V\vert\)iff there exists a Hamiltonian Cycle in original graph.

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