Euclidean Tsp (part I)

These notes are based on scribe notes by Marios Papaefthymiou and Mike Kluger-man. 1 Euclidean TSP Consider the travelling salesman problem in the plane. Given n points in the plane, we would like to nd a tour that visits all of them and that minimizes the distance travelled, where the distance between two points is given by the Euclidean distance. This problem will be denoted by ETSP (Euclidean TSP). In a companion set of notes, we will present very recent algorithms of Arora and Mitchell that produce in polynomial time a tour which is within 1+ of the optimum, for any xed > 0. In this set of notes, we rst present some preliminaries and older related results. Before we continue, we should point out that it is not known whether the Euclidean TSP is in NP. Even if we are presented with a candidate tour for the \yes-no" version of the problem, we do not know how to avoid computing a possibly exponential number of decimal digits, in order to calculate the square root required for the Euclidean distance. First, we present an algorithm which generates a path of length no more than p N k (k a constant to be determined) given that the points all lie within a unit square. Here we use the \Strips Method". First, break the square down into p N c horizontal strips of equal height. The \Salesman's" strategy will then be the following: He will begin at the left side of the topmost strip and travel to the right along the horizontal line splitting the strip in half. If at any point the salesman reaches a spot where a point, p, in the problem is located in the strip directly above or below him, he travels directly to p, and back to the center line. When he reaches the end of the line, he travels along the edge of the square to the middle of the next strip and then travels left across the middle line. The salesman goes back and forth in this way until he has passed through all strips. At the end, the salesman travels from the lower right corner to the upper left to nish the loop (see Figure 1, part a). Analysis: Let 1. A = length travelled horizontally across each strip = 1. 2. B = distance travelled vertically along edge of square. 3. C = …