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Author: Admin | 2025-04-28
Or distance of that road), find a Hamiltonian cycle with the least weight. This is more general than the Hamiltonian path problem, which only asks if a Hamiltonian path (or cycle) exists in a non-complete unweighted graph.The requirement of returning to the starting city does not change the computational complexity of the problem; see Hamiltonian path problem.Another related problem is the bottleneck travelling salesman problem: Find a Hamiltonian cycle in a weighted graph with the minimal weight of the weightiest edge. A real-world example is avoiding narrow streets with big buses.[15] The problem is of considerable practical importance, apart from evident transportation and logistics areas. A classic example is in printed circuit manufacturing: scheduling of a route of the drill machine to drill holes in a PCB. In robotic machining or drilling applications, the "cities" are parts to machine or holes (of different sizes) to drill, and the "cost of travel" includes time for retooling the robot (single-machine job sequencing problem).[16]The generalized travelling salesman problem, also known as the "travelling politician problem", deals with "states" that have (one or more) "cities", and the salesman must visit exactly one city from each state. One application is encountered in ordering a solution to the cutting stock problem in order to minimize knife changes. Another is concerned with drilling in semiconductor manufacturing; see e.g., U.S. patent 7,054,798. Noon and Bean demonstrated that the generalized travelling salesman problem can be transformed into a standard TSP with the same number of cities, but a modified distance matrix.The sequential ordering problem deals with the problem of visiting a set of cities, where precedence relations between the cities exist.A common interview question at Google is how to route data among data processing nodes; routes vary by time to transfer the data, but nodes also differ by their computing power and storage, compounding the problem of where to send data.The travelling purchaser problem deals with a purchaser who is charged with purchasing a set of products. He can purchase these products in several cities, but at different prices, and not all cities offer the same products. The objective is to find a route between a subset of the cities that minimizes total cost (travel cost + purchasing cost) and enables the purchase of all required products.Integer linear programming formulations[edit]The TSP can be formulated as an integer linear program.[17][18][19] Several formulations are known. Two notable formulations are the Miller–Tucker–Zemlin (MTZ) formulation and the Dantzig–Fulkerson–Johnson (DFJ) formulation. The DFJ formulation is stronger, though the MTZ formulation is still useful in certain settings.[20][21]Common to both these formulations is that one labels the cities with the numbers and takes 0}"> to be the cost (distance) from city to city . The main variables in the formulations are:It is because these are 0/1 variables that the formulations become integer programs; all other constraints are purely linear. In particular, the objective in the program is to minimize the tour lengthWithout further constraints, the will effectively range over all subsets of the set of edges,
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