Graph is a complete graph -it called K6- IT HAS 6 VETICES. also, every vertex is connected to every other vertex.
a) how many possible Hamilton circuits ( say starting from A) including reversals, does the graph have?
b) if we were to use the Brute Force algorithm to solve the Traveling Salesman problem on this graph, how many circuits ( say starting from A) would list, including reversals?
c) how many circuits would we list if we used the Repetitive Nearest Algorithm to find an approximate solution to the TSP?
a) As the complete graph K6 has 6 vertices, there are 5 possible vertices to visit next after starting from A. After visiting one of these vertices, there are 4 remaining to choose from, then 3, then 2, and finally, 1. Therefore, there are 5 * 4 * 3 * 2 * 1 = 120 possible Hamilton circuits starting from A, including reversals.
b) In the Brute Force algorithm, we would need to consider all possible permutations of the 6 vertices. The number of permutations of 6 items is 6! (6 factorial), which is equal to 720. However, since the starting vertex is fixed at A, we divide this by 6 (the number of starting positions) to avoid duplicate circuits. So, there would be 720 / 6 = 120 circuits listed, including reversals.
c) The Repetitive Nearest Algorithm is an approximate solution, and the number of circuits it would list depends on the starting vertex chosen and the iterations performed. However, in general, the Repetitive Nearest Algorithm would significantly reduce the number of circuits explored compared to the Brute Force algorithm. The exact number would depend on the specific implementation and the choices made during the algorithm's execution.
a) To find the number of possible Hamilton circuits in a complete graph with 6 vertices (K6), we can start at any vertex and then visit the remaining 5 vertices in any order. Therefore, the number of possible circuits, including reversals, is (6-1)!, which is equal to 5!.
b) The Brute Force algorithm for solving the Traveling Salesman Problem involves listing all possible permutations of the vertices and calculating the total distance for each permutation. Since there are 6 vertices in the graph, we would have 6! possible circuits. However, since reversals are also included, the total number of circuits, including reversals, would be 2 times 6!. Therefore, the number of circuits, including reversals, would be 2 * 6!.
c) The Repetitive Nearest Algorithm is an approximation algorithm for solving the Traveling Salesman Problem. The algorithm starts with an arbitrary vertex and repeatedly selects the nearest unvisited vertex until all vertices are visited. In this case, with 6 vertices, the Repetitive Nearest Algorithm would produce a single circuit, starting from vertex A, by visiting all 6 vertices in a specific order. Therefore, the number of circuits obtained using this algorithm would be 1.
a) To find the number of possible Hamilton circuits in a complete graph with 6 vertices, including reversals, we need to determine the number of different permutations of the vertices when starting from vertex A.
In this case, we have 6 vertices in total. If we fix vertex A as the starting point, we have 5 remaining vertices to visit. We can select the next vertex from the remaining 5 vertices, then the next from the remaining 4, and so on, until all vertices are visited.
Therefore, the number of possible Hamilton circuits, including reversals, is (5!) * 2, which is 120 * 2 = 240.
b) The Brute Force algorithm considers all possible permutations and computes the total cost of each permutation for the Traveling Salesman Problem (TSP). For a graph with 6 vertices, there are 6! (720) possible permutations.
Since we are starting from vertex A, we need to consider that each permutation can be reversed. Therefore, the total number of circuits, including reversals, for the Brute Force algorithm would be 720 * 2 = 1440.
c) The Repetitive Nearest Neighbor Algorithm is an approximation algorithm for the TSP. It starts at a random vertex and repeatedly chooses the nearest unvisited vertex until all vertices have been visited.
To find the number of circuits when using this algorithm, we can consider the number of possible starting vertices and multiply it by the number of possible reversals.
There are 6 possible starting vertices, and each starting vertex can be reversed. Therefore, the total number of circuits, including reversals, when using the Repetitive Nearest Neighbor Algorithm would be 6 * 2 = 12.