Which of these problems can be reduced to the Traveling Salesman Problem? Hamiltonian Cycle 2-SAT Hamiltonian Path 3-SAT
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In a country, there are N cities, and there are some undirected roads between them. For every city there is an integer value (may be positive, negative, or zero) on it. You want to know, if there exists a cycle (the cycle cannot visit a city or a road twice), and the sum of values of the cities on the cycle is equal to 0. Note, a single vertex is not a cycle. Prove this problem is NP-Complete. Use a reduction from the Hamiltonian cycle problem. Prove the claim in the direction from the Hamiltonian cycle problem to the reduced problem.
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Let FHAM be the problem of finding a Hamiltonian cycle in a graph $G$ and DHAM be the problem of determining if a Hamiltonian cycle exists in a graph. Which one of the following is true? (A) Both FHAM and DHAM are NP-hard. (B) FHAM is NP-hard, but DHAM is not. (C) DHAM is NP-hard but FHAM is not. (D) Neither DHAM nor FHAM is NP-hard.
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Solve the traveling salesperson problem for this graph by finding the total weight of all Hamilton circuits and determining a circuit with minimum total weight.
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