Unformatted text preview: 5 of cities with �S� �
n, there must be some city not in 5 that follows some city that is in 5. This set of constraints is known
as the subtour elimination Constraints.
With all of the constraints listed above, every feasible solution corresponds to a tour.
Implementation details.
Suppose that one wanted to solve the linear programming relaxation of the TSP� that is, we solve the
problem obtained from the above constraints if we drop the integrality constraints, and merely
require that xij � 0 for each i and j. (We wonCt deal with solving the integer program here.) We refer
to this linear program as LP�.
At first it appears that there is no way of solving LP� for any large values of n, say n > 100. For any
TSP instance with more than 100 cities, there are more than 2100 different subtour elimination
constraints. Listing all of the constraints would take more than all of the computer memory in the
entire world.
Instead, LP� is solved using a "constraint generation approach." LP(0) is obtained from LP� by
dropping all of the subtour elimination constraints. An optimal solution x0 is obtained for LP(0). If
x0 is feasible for LP�, then it is also optimal for LP�. If not, then some subtour e...
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 Spring '07
 JamesOrli
 Management

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