If we can solve the Hamiltonian path in time $O(n^4)$ then you can solve any other NPC problem in $O(n^4)$ time. Is it true of false? I think it is false, even tho Hamiltonian path problem in NPC it doesnt mean that all NPC problems will be solved in the same time.
2026-03-27 05:34:55.1774589695
Hamiltonian path problem vs other NPC problems
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You are correct. There might be an $NP$-complete problem that can only be translated into Hamiltonian path problem in $\Omega(n^{10})$ time, for instance. In which case an $\Omega(n^{10})$ translation and an $O(n^4)$ solution gives a total time of $\Omega(n^{10})$. The only thing you can be certain of is that the final time is bounded by a polynomial.