Why aren't these 2 problems, namely TSP and Hamiltonian path problem , both NP-complete?
They seem identical.
For a problem X to be NP-complete , it has to satisfy:
There are two versions of the The Travelling Salesman Problem (TSP) :
The definitions of NP-hardness and NP-completeness are related but different. Specifically, a problem is NP-hard if every problem in NP reduces to it in polynomial time, and a problem is NP-complete if it's both NP-hard and itself in NP.
The class NP consists of decision problems, problems that have a yes/no answer. As a result, TSP cannot be in NP because the expected answer is a number rather than yes or no. Therefore, TSP can be NP-hard, but it can't be NP-complete.
On the other hand, the Hamiltonian path problem asks for a yes/no answer, and it happens to be in NP. Therefore, since it's NP-hard as well, it's NP-complete.
Now, you can take TSP and convert it to a decision problem by changing the question from "what's the cheapest path?" to "is there a path that costs X or less?," and that latter formulation is in NP and also happens to be NP-complete.
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