We consider the traveling salesman problem when the cities are points in Rd for some fixed d and distances are computed according to a polyhedral norm. We show that for any such norm, the problem of finding a tour of maximum length can be solved in polynomial time. If arithmetic operations are assumed to take unit time, our algorithms run in time O(n f-2 log n), where f is the number of facets of the polyhedron determining the polyhedral norm. Thus for example we have O(n 2 log n) algorithms for the cases of points in the plane under the Rectilinear and Sup norms. This is in contrast to the fact that finding a minimum length tour in each case is NP-hard.
|Title of host publication||Integer Programming and Combinatorial Optimization (Proceedings 6th International IPCO Conference, Houston TX, USA, June 22-24, 1998)|
|Editors||R.E. Bixby, E.A. Boyd, R.Z. Rios-Mercado|
|Place of Publication||Berlin|
|Publication status||Published - 1998|
|Name||Lecture Notes in Computer Science|