Polynomial time algorithms for maximization problems in spaces with polyhedral norms

We discuss geometric special cases of optimization problems on graphs where the n vertices are represented as points in IR d and where the lengths of the edges are measured according to a norm with a polyhedral unit ball. We derive algorithms whose running time is polynomial in n (and depends exponentially on the number of faces of the unit ball). We consider the maximum travelling salesman problem, the weighted maximum 3-dimensional matching problem, and the weighted maximum 3-dimensional assignment problem. One consequence of our results is an O(n 13) polynomial time algorithm for the Maximum TSP on n cities in IR 2 under Manhattan distances. The computational complexity of the Maximum TSP with Euclidean distances, however, remains unsettled.