Two solvable cases of the Traveling Salesman Problem

In the Euclidean Traveling Salesman Problem, a set of points in the plane is given, and we look for a shortest closed curve through these lines (a \tour"). We treat two special cases of this problem which are solvable in polynomial time. The rst solvable case is the N-line Traveling Salesman Problem, where the points lie on a small number (N) of parallel lines. Such problems arise for example in the fabrication of printed circuit boards, where the distance traveled by a laser which drills holes in certain places of the board should be minimized. By a dynamic programming algorithm, we can solve the N-line Traveling Salesman Problem for n points in time nN , for xed N , i. e., in polynomial time. This extends a result of Cutler (1980) for 3 lines. The parallelity condition can be relaxed to point sets which lie on N \almost parallel" line segments. The other solvable case concerns the necklace condition: A tour is a necklace tour if we can draw disks with the given points as centers such that two disks intersect if and only if the corresponding points are adjacent on the tour. If a necklace tour exists, it is the unique optimal tour. We give an algorithm which tests in O(n3=2) time whether a given tour is a necklace tour, improving an algorithm of Edelsbrunner, Rote, and Welzl (1988) which takes O(n2) time. It is based on solving a system of linear inequalities by the generalized nested dissection procedure of Lipton, Rose, and Tarjan (1979). We describe how this method can be implemented with only linear storage. We give another algorithm which tests in O(n2 logn) time and linear space, whether a necklace tour exists for a given point set, by transforming the problem to a fractional 2-factor problem on a sparse bipartite graph. Both algorithms also compute radii for the disks realizing the necklace tour.