The longest-simple-cycle problem is the problem of determining a simple cycle(no repeated vertices) of maximum length in a graph. Formulate a related decisionproblem, and show that the decision problem is NP-complete.
Question
The longest-simple-cycle problem is the problem of determining a simple cycle(no repeated vertices) of maximum length in a graph. Formulate a related decisionproblem, and show that the decision problem is NP-complete.
Solution
The related decision problem to the longest-simple-cycle problem could be formulated as follows:
"Given a graph G and an integer k, does there exist a simple cycle (no repeated vertices) in G of length at least k?"
To show that this decision problem is NP-complete, we need to prove two things:
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The problem is in NP: We can easily verify a solution in polynomial time. Given a cycle, we can check if it is simple (no repeated vertices) and if its length is at least k in polynomial time.
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The problem is NP-hard: This means that every problem in NP can be reduced to this problem in polynomial time. The Hamiltonian cycle problem, which is known to be NP-complete, can be reduced to this problem. Given an instance of the Hamiltonian cycle problem, we can create an instance of our problem by setting k to be the number of vertices in the graph. If there is a Hamiltonian cycle in the graph, then there is a simple cycle of length at least k. Conversely, if there is a simple cycle of length at least k, then there is a Hamiltonian cycle in the graph. This reduction can be done in polynomial time.
Therefore, the decision problem is NP-complete.
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