Travelling Sales Person ProblemThe traveling salesman problems abide by a salesman and a set of cities. The salesman has to visit every one of the cities starting from a certain one (e.g., the hometown) and to return to the same city. The challenge of the problem is that the traveling salesman needs to minimize the total length of the trip. Suppose the cities are x_{1} x_{2}..... x_{n} where cost c_{ij} denotes the cost of travelling from city x_{i} to x_{j}. The travelling salesperson problem is to find a route starting and ending at x_{1} that will take in all cities with the minimum cost. Example: A newspaper agent daily drops the newspaper to the area assigned in such a manner that he has to cover all the houses in the respective area with minimum travel cost. Compute the minimum travel cost. The area assigned to the agent where he has to drop the newspaper is shown in fig: Solution: The cost adjacency matrix of graph G is as follows: cost_{ij} = The tour starts from area H_{1} and then select the minimum cost area reachable from H_{1}. Mark area H_{6} because it is the minimum cost area reachable from H_{1} and then select minimum cost area reachable from H_{6}. Mark area H_{7} because it is the minimum cost area reachable from H_{6} and then select minimum cost area reachable from H_{7}. Mark area H_{8} because it is the minimum cost area reachable from H_{8}. Mark area H_{5} because it is the minimum cost area reachable from H_{5}. Mark area H_{2} because it is the minimum cost area reachable from H_{2}. Mark area H_{3} because it is the minimum cost area reachable from H_{3}. Mark area H_{4} and then select the minimum cost area reachable from H_{4} it is H_{1}.So, using the greedy strategy, we get the following. 4 3 2 4 3 2 1 6 H_{1} → H_{6} → H_{7} → H_{8} → H_{5} → H_{2} → H_{3} → H_{4} → _{H1}. Thus the minimum travel cost = 4 + 3 + 2 + 4 + 3 + 2 + 1 + 6 = 25 Matroids:A matroid is an ordered pair M(S, I) satisfying the following conditions:
We say that a matroid M (S, I) is weighted if there is an associated weight function w that assigns a strictly positive weight w (x) to each element x ∈ S. The weight function w extends to a subset of S by summation: w (A) = ∑_{x∈A} w(x) for any A ∈ S.
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