Annals of Operations Research

, Volume 86, Issue 0, pp 529–558 | Cite as

New classes of efficiently solvable generalized Traveling Salesman Problems

  • E. Balas


We consider the n‐city traveling salesman problem (TSP), symmetric or asymmetric,with the following attributes. In one case, a positive integer k and an ordering (1,..., n) ofthe cities is given, and an optimal tour is sought subject to the condition that for any pairi, j ∈ (1..., n), if j ≥ i + k, then i precedes j in the tour. In another case, position i in the tourhas to be assigned to some city within k positions from i in the above ordering. This case isclosely related to the TSP with time windows. In a third case, an optimal tour visiting m outof n cities is sought subject to constraints of the above two types. This is a special case ofthe Prize Collecting TSP (PCTSP). In any of the three cases, k may be replaced by city‐specificintegers k(i), i = 1,..., n. These problems arise in practice. For each class, we reducethe problem to that of finding a shortest source‐sink path in a layered network with a numberof arcs linear in n and exponential in the parameter k (which is independent of the problemsize). Besides providing linear time algorithms for the solution of these problems, the reductionto a shortest path problem also provides a compact linear programming formulation.Finally, for TSPs or PCTSPs that do not have the required attributes, these algorithms canbe used as heuristics that find in linear time a local optimum over an exponential‐sizeneighborhood.


Positive Integer Time Window Short Path Local Optimum Linear Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Kluwer Academic Publishers 1999

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  • E. Balas

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