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travelling_salesman_problem

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travelling_salesman_problem [2015/09/22 12:49] bsuresh Minor change, added \ldots |
travelling_salesman_problem [2018/10/01 13:29] (current) bsuresh old revision restored (2017/04/05 23:01) |
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====== Travelling salesman problem ====== | ====== Travelling salesman problem ====== | ||

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+ | <note>This is a playground page for new users. Feel free to edit here!</note> | ||

The travelling salesman problem (TSP) asks the following question: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science. | The travelling salesman problem (TSP) asks the following question: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science. | ||

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===== Integer linear programming formulation ===== | ===== Integer linear programming formulation ===== | ||

- | TSP can be formulated as an integer linear program. Label the cities with the numbers $0, \ldots, n$ and define: | + | TSP can be formulated as an integer linear program. Label the cities with the numbers $0, ..., n$ and define: |

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x_{ij} = \begin{cases} 1 & \text{the path goes from city } i \text{ to city } j \\ 0 & \text{otherwise} \end{cases} | x_{ij} = \begin{cases} 1 & \text{the path goes from city } i \text{ to city } j \\ 0 & \text{otherwise} \end{cases} |

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