Homework 12
Last updated: Thu, 5 May 2022 12:50:26 -0400
Out: Wed May 04, 00:00 EST Due: Wed May 11, 23:59 EST
This assignment continues to explore NP-Completeness and NP-Complete problems.
Homework Problems
Undirected Hamiltonian Paths (12 pts)
Hamiltonian Cycles (12 pts)
Making Hamiltonian Paths (11 pts)
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 \textit{UHAMPATH} (from lecture) is \textbf{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 \textbf{NP}-Complete.
\textit{HCYCLE} = \left\{G\mid G\textrm{ is a directed graph with a Hamiltonian cycle}\right\}
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 \textbf{NP}-Complete.
\textit{HMAKE} = \left\{\left\langle G,k\right\rangle\mid G\textrm{ is a directed graph that has a Hamiltonian path if } k\textrm{ edges are added to it}\right\}
Make sure to include all the required parts of the proof.