- #1
neurocomp2003
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sup, just realized there is not graphtheory/combinatorial subforums.
is the hamiltonian cycle still considered and NP problem?
is the hamiltonian cycle still considered and NP problem?
A Hamiltonian is a mathematical function that describes the total energy of a system. It is used in physics and engineering to model and analyze the behavior of dynamic systems. In computer science, the term Hamiltonian is often used in relation to the complexity class NP.
In computational complexity theory, NP (nondeterministic polynomial time) is a class of decision problems that can be solved by a nondeterministic Turing machine in polynomial time. A Hamiltonian being in NP means that there exists a polynomial-time algorithm to solve the problem of finding a Hamiltonian cycle in a graph.
No, it is not. In the 1970s, it was proven that the problem of finding a Hamiltonian cycle in a graph is NP-complete. However, in 2002, it was shown that the Hamiltonian path problem is NP-complete, but the Hamiltonian cycle problem is not. This means that the problem of determining whether a graph has a Hamiltonian cycle is in the complexity class NP, but it is not NP-complete.
It is not known whether the Hamiltonian cycle problem can be solved in polynomial time. It is an open problem in computer science and mathematics. Some researchers believe that there is no polynomial-time algorithm for this problem, while others believe that there may be a more efficient algorithm yet to be discovered.
The question is important because the complexity class NP is a fundamental concept in computer science and mathematical theory. Understanding the complexity of the Hamiltonian cycle problem can provide insights into the complexity of other problems in NP and help researchers develop more efficient algorithms for solving them. It also has practical applications in areas such as network optimization and logistics planning.