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# How do you prove that the Hamiltonian path is NP-complete?

## How do you prove that the Hamiltonian path is NP-complete?

Any Hamiltonian Path can be made into a Hamiltonian Circuit through a polynomial time reduction by simply adding one edge between the first and last point in the path. Therefore we have a reduction, which means that Hamiltonian Paths are in NP Hard, and therefore in NP Complete.

### Why Hamiltonian problem is NP-complete?

Thus we can say that the graph G’ contains a Hamiltonian Cycle iff graph G contains a Hamiltonian Path. Therefore, any instance of the Hamiltonian Cycle problem can be reduced to an instance of the Hamiltonian Path problem. Thus, the Hamiltonian Cycle is NP-Hard.

How do you solve a Hamiltonian path problem?

Simple way of solving the Hamiltonian Path problem would be to permutate all possible paths and see if edges exist on all the adjacent nodes in the permutation. If the graph is a complete graph, then naturally all generated permutations would quality as a Hamiltonian path.

Is Hampath NP-complete?

We have to show Hamiltonian Path is NP-Complete. Hamiltonian Path or HAMPATH in a directed graph G is a directed path that goes through each node exactly once. Here the certificate will be a Hamiltonian path from s to t itself in G if exists. So HAMPATH is in NP proved.

## What if P is not equal to NP?

If P equals NP, every NP problem would contain a hidden shortcut, allowing computers to quickly find perfect solutions to them. But if P does not equal NP, then no such shortcuts exist, and computers’ problem-solving powers will remain fundamentally and permanently limited.

### Is Travelling salesman problem NP-complete?

Traveling Salesman Optimization(TSP-OPT) is a NP-hard problem and Traveling Salesman Search(TSP) is NP-complete. However, TSP-OPT can be reduced to TSP since if TSP can be solved in polynomial time, then so can TSP-OPT(1).

Is a Hamiltonian path a cycle?

A Hamiltonian path or traceable path is a path that visits each vertex of the graph exactly once. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph.

Is halting a problem with NP?

It is also easy to see that the halting problem is not in NP since all problems in NP are decidable in a finite number of operations, but the halting problem, in general, is undecidable. There are also NP-hard problems that are neither NP-complete nor Undecidable.

## Is the problem of finding a Hamiltonian path in FNP?

The problem of finding a Hamiltonian cycle or path is in FNP; the analogous decision problem is to test whether a Hamiltonian cycle or path exists. The directed and undirected Hamiltonian cycle problems were two of Karp’s 21 NP-complete problems. They remain NP-complete even for special kinds of graphs, such as:

### Is the Hamiltonian cycle a NP hard problem?

Thus we can say that the graph G’ contains a Hamiltonian Cycle iff graph G contains a Hamiltonian Path. Therefore, any instance of the Hamiltonian Cycle problem can be reduced to an instance of the Hamiltonian Path problem. Thus, the Hamiltonian Cycle is NP-Hard. Conclusion: Since, the Hamiltonian Cycle is both, a NP-Problem and NP-Hard.

Can a directed graph have a Hamiltonian path?

We have to show Hamiltonian Path is NP-Complete. Hamiltonian Path or HAMPATH in a directed graph G is a directed path that goes through each node exactly once. We Consider the problem of testing whether a directed graph contain a Hamiltonian path connecting two specified nodes, i.e.

Is there a proof that hampath is NP complete?

In either case path cannot enter from because is only available node that points at, so path must exit via . Hence Hamiltonian path must be normal. This reduction obviously operates in polynomial time and hence the proof is complete that HAMPATH is NP-Complete.