This assignment continues to explore NP-Completeness and NP-Complete problems.

Homework Problems

1. Undirected Hamiltonian Paths (12 pts)

2. Hamiltonian Cycles (12 pts)

3. Making Hamiltonian Paths (11 pts)

4. README (1 point)

Total: 36 points

Submitting

Submit your solution to this assignment in Gradescope hw12. Please assign each page to the correct problem and make sure your solutions are legible.

A submission must also include a README containing the required information.

1 Undirected Hamiltonian Paths

Prove that UHAMPATH (from lecture) is NP-Complete. Start with the ideas from class. Make sure to include all the required parts of the proof as described in lecture.

2 Hamiltonian Cycles

Recall that a cycle in a graph (see Sipser Ch 0) is a path that starts and ends at the same vertex. Also, a Hamiltonian path is a path that touches every vertex in the graph.

Prove that the following language is NP-Complete.

HCYCLE={G|G is a directed graph with a Hamiltonian cycle}

Make sure to include all the required parts of the proof.

3 Making Hamiltonian Paths

Recall that a Hamiltonian path is a path that touches every vertex in the graph.

Prove that the following language is NP-Complete.

HMAKE={?G,k?|G is a directed graph that has a Hamiltonian path if k edges are added to it}

Make sure to include all the required parts of the proof.

Homework Problems

1. Undirected Hamiltonian Paths (12 pts)

2. Hamiltonian Cycles (12 pts)

3. Making Hamiltonian Paths (11 pts)

4. README (1 point)

Total: 36 points

Submitting

Submit your solution to this assignment in Gradescope hw12. Please assign each page to the correct problem and make sure your solutions are legible.

A submission must also include a README containing the required information.

1 Undirected Hamiltonian Paths

Prove that UHAMPATH (from lecture) is NP-Complete. Start with the ideas from class. Make sure to include all the required parts of the proof as described in lecture.

2 Hamiltonian Cycles

Recall that a cycle in a graph (see Sipser Ch 0) is a path that starts and ends at the same vertex. Also, a Hamiltonian path is a path that touches every vertex in the graph.

Prove that the following language is NP-Complete.

HCYCLE={G|G is a directed graph with a Hamiltonian cycle}

Make sure to include all the required parts of the proof.

3 Making Hamiltonian Paths

Recall that a Hamiltonian path is a path that touches every vertex in the graph.

Prove that the following language is NP-Complete.

HMAKE={?G,k?|G is a directed graph that has a Hamiltonian path if k edges are added to it}

Make sure to include all the required parts of the proof.

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