Theorem 36.15: 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 TSP
Given graph G = (V, E), transformation f outputs complete graph with vertices V.
Weights of edges = 1 if , or (|V| + 1) if
Also outputs the number |V|.
f is clearly implementable in polynomial time.

4.

Then there exists a tour in this complete graph of size iff there exists a Hamiltonian Cycle in original graph.

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